Displacement measurement method and apparatus, strain measurement method and apparatus, elasticity and visco-elasticity constants measurement apparatus, and the elasticity and visco-elasticity constants measurement apparatus-based treatment apparatus

ABSTRACT

The present invention provides elastic constant and visco elastic constant measurement apparatus etc. for measuring in the ROI in living tissues elastic constants such as shear modulus, Poisson&#39;s ratio, Lame constants, etc., visco elastic constants such as visco shear modulus, visco Poisson&#39;s ratio, visco Lame constants, etc. and density even if there exist another mechanical sources and uncontrollable mechanical sources in the object. The elastic constant and visco elastic constant measurement apparatus is equipped with means of data storing  2  (storage of deformation data measured in the ROI  7  etc.) and means of calculating elastic and visco elastic constants  1  (calculator of shear modulus etc. at arbitrary point in the ROI from measured strain tensor data etc.), the means of calculating elastic and visco elastic constants numerically determines elastic constants etc. from the first order partial differential equations relating elastic constants etc. and strain tensor etc.

This is a divisional of application Ser. No. 10/326,526 filed Dec. 23,2002. The entire disclosure of the prior application, application Ser.No. 10/326,526 is considered part of the disclosure of the accompanyingDivisional application and is hereby incorporated by reference.

This invention claims priority to prior Japanese Patent Applications2001-389484, filed Dec. 21, 2001, 2002-10368, filed Jan. 18, 2002,JP2002-123765, filed Apr. 25, 2002, JP2002-143145, filed May 17, 2002,JP2002-176242, filed Jun. 17, 2002, JP2002-178265, filed Jun. 19, 2002,JP2002-238434, filed Aug. 19, 2002, JP2002-240339, filed Aug. 21, 2002,JP2002-241647, filed Aug. 22, 2002, and JP2002-256807, filed Sep. 2,2002, the disclosures of which all are incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an apparatus and method forlow-destructively (low-invasively) measuring mechanical propertieswithin object such as structures, substances, materials, living tissues(liver, prostate, breast, etc.). For instance, measured can be, due toapplied stress and/or vibration by arbitrary mechanical sources,generated displacement vector, strain tensor, strain rate tensor,acceleration vector, or velocity vector, etc. within the body.Furthermore, from these measured deformation data, following constantscan be measured, elastic constants such as shear modulus, Poisson'sratio, etc., visoc elastic constants such as visco shear modulus, viscoPoisson's ratio, etc., delay times or relaxation times relating theseelastic constants and visco elastic constants, or density.

On typical applied field, i.e., medical field, such as ultrasonicdiagnosis, nuclear magnetic resonance diagnosis, light diagnosis, radiotherapeutics, the present method and apparatus can be applied formonitoring tissue degeneration, i.e., treatment effectiveness.Otherwise, on structures, substances, materials, living tissues,measured static and/or dynamic mechanical properties can be utilized forevaluation, examination, diagnosis, etc.

2. Description of a Related Art

For instance, on medical field (liver, prostate, breast, etc.), lesionsare proposed to be treated by cryotherapy, or by applying radioactiveray, high intensity focus ultrasound, laser, electromagnetic RF wave,micro wave, etc. In these cases, the treatment effectiveness is proposedto be monitored. Moreover, chemotherapy (anti-cancer drag etc.)effectiveness is also proposed to be monitored. For instance, onradiotherapy etc., the treatment effectiveness can be confirmed bylow-invasively measuring degeneration (including temperature change) ofthe lesion. Otherwise, due to applied stress to the tissue part ofinterest including lesions, generated deformations and deformationchanges are measured, from which the pathological state of the tissue isevaluated such as elastic constants etc. Thus, based on the measureddistinct pathological state, the part of interest is diagnosed, ortreatment effectiveness is observed.

Temperature is known to have high correlations with elastic constants,visco elastic constants, delay times or relaxation times relatingelastic constants and visco elastic constants, and density, etc.Therefore, by measuring the following constants, temperaturedistribution can be measured, i.e., elastic constants such as shearmodulus, Poisson's ratio, etc., visco elastic constants such as viscoshear modulus, visco Poisson's ratio, etc., delay times or relaxationtimes relating these elastic constants and visco elastic constants, ordensity.

In the past, elastic constants and visco elastic constants have beenmeasured by applying stress at many points and by measuring theresponses such as stresses and strains. That is, stress meter and strainmeter are used, and sensitivity analysis is numerically performed withutilization of finite difference method or finite element method.Otherwise, in addition to elastic constants, visco elastic constantssuch as visco shear modulus, visco Poisson's ratio, etc. has been alsomeasured by estimating the shear wave propagation velocity generated byapplying vibrations.

As other monitoring techniques, temperature distribution is measured byevaluating nuclear magnetic resonance frequency, electric impedance,ultrasound velocity, etc. However, to measure temperature, thesetechniques need other relating physical properties of the target tissue.If degeneration occurs on the part of region, the relating physicalproperties also change; thus resulting severe limitations of thetemperature measurement.

Other disadvantage is that the past measurement technique needs manyindependent deformation fields generated by mechanical sources outsidethe target body. However, if there exist internal mechanical sources ormechanical sources are uncontrollable, the technique becomesunavailable. That is, the past technique needs all information aboutmechanical sources, such as positions, force directions, forcemagnitudes, etc. Moreover, the technique needs stress data and straindata at the target body surface, and needs whole body model (finitedifference method or finite element method). Furthermore, low arespatial resolutions of measured elastic constants and visco elasticconstants from shear wave velocity.

On the other hand, medical ultrasound diagnosis apparatus canlow-invasively image tissue distribution by converting ultrasonic echosignals (echo signals) to image, after transmitting ultrasonic pulses totarget tissue and receiving the echo signals at ultrasound transducer.Thus, by ultrasonically measuring tissue displacements generated due toarbitrary mechanical sources, or by measuring generated tissue strains,tissue elastic constants, etc., differences of between lesion and normaltissue can be observed low-invasively. For instance, measured within thebody can be, due to applied stress and/or vibration by arbitrarymechanical sources, generated displacement vector, strain tensor, strainrate tensor, acceleration vector, velocity vector, etc. Furthermore,from these measured deformation data, following constants can bemeasured, elastic constants such as shear modulus, Poisson's ratio,etc., visco elastic constants such as visco shear modulus, viscoPoisson's ratio, etc., delay times or relaxation times relating theseelastic constants and visco elastic constants, or density.

Then, in the past, tissue displacement has been proposed to be measuredto low-invasively diagnose tissue and lesion by evaluating the echosignal changes of more than one time transmitting signal. From themeasured displacement distribution, strain distribution is obtained, bywhich distribution of pathological state of tissue has been proposed tobe diagnosed (Japanese Patent Application Publication JP-A-7-55775,JP-A-2001-518342). Specifically, 3 dimensional (3D), 2D, or 1D region ofinterest (ROI) are set in the target body, and distributions of three,two, or one displacement component are measured, from which in additionto strain tensor distribution, elastic constant distributions, etc. arealso numerically obtained.

In addition to ultrasound transducer, as the displacement (strain)sensor, utilized can be known contact or non-contact sensors such aselectromagnetic wave (including light) detector etc. As mechanicalsources, compressor and vibrator can be, transducer-mounted apparatuses,not transducer-mounted ones, internal heart motion, respiratory motion.If ROI is deformed by ultrasound transmitted from sensor, there may notneed other mechanical sources except for the sensor. In addition to thestationary elastic constants, difference of the tissue pathologicalstate includes dynamic changes of elastic constants and temperature dueto treatment.

However, as the classical tissue displacement measurement methods assumetissue being deformed only in the axial (beam) direction, when tissuealso moves in lateral (scan) direction, the classical method has lowaxial measurement accuracy. That is, the displacement was determinedonly by 1D axial processing of ultrasound echo signals (hereafter, echosignal includes rf echo signal, quadrate detection signal, envelopdetection signal, and complex signal).

Recently, displacement accuracy is improved by us through development of2D displacement vector measurement method, i.e., phase gradientestimation method of the 2D echo cross-spectrum based on so-called the2D cross-correlation processing and the least squares processing. Thismethod can suitably cope with internal, uncontrollable mechanicalsources (e.g., heart motion, respiratory motion, blood vessel motion,body motion, etc.).

However, strictly speaking, measurement accuracy of actual 3D tissuedisplacement becomes low because the method can measure by 2D processingof echo signals two displacement components or by 1D processing onedisplacement component.

Particularly, as lateral component of echo signal has a narrow bandwidthand has no carrier frequency, lateral displacement measurement accuracyand spatial resolution are much lower compared with axial ones. Thus,the low lateral measurement accuracy degrades the 3D displacement vectormeasurement and the 3D strain tensor measurement.

Furthermore, when large displacement needs to be handled, beforeestimating the gradient of the cross-spectrum phase, the phase must beunwrapped, or the displacement must be coarsely estimated bycross-correlation method as multiples of sampling intervals. Thus,measurement process had become complex one.

SUMMARY OF THE INVENTION

The first purpose of the present invention is to provide an apparatusand method for low-destructively measuring mechanical properties withinobject such as structures, substances, materials, living tissues (liver,prostate, breast, etc.), even if there exists internal, oruncontrollable mechanical sources. The first purpose of the presentinvention is, for instance, for diagnosing and monitoring treatmenteffectiveness on living tissue, to provide the measurement technique offollowing constants, elastic constants such as shear modulus, Poisson'sratio, etc., visco elastic constants such as visco shear modulus, viscoPoisson's ratio, etc., delay times or relaxation times relating theseelastic constants and visco elastic constants, or density.

The second purpose of the present invention is to provide thelow-invasive treatment technique with utilization of low-invasivemeasurement of the following constants, elastic constants such as shearmodulus, Poisson's ratio, etc., visco elastic constants such as viscoshear modulus, visco Poisson's ratio, etc., delay times or relaxationtimes relating these elastic constants and visco elastic constants, ordensity.

The third purpose of the present invention is to improve measurementaccuracy of displacement vector distribution generated in 3D, 2D(including or not including beam direction), or 1D (beam direction orscan direction) ROI in the target body when estimating gradient of theecho cross-spectrum phase. Cross-spectrum can be also estimated byFourier's transform of echo cross-correlation function.

The fourth purpose of the present invention is to simplify calculationprocess into one without unwrapping the cross-spectrum phase norutilizing cross-correlation method; thus reducing calculation amount andshortening calculation time.

The fifth purpose of the present invention is to improve measurementaccuracy of lateral displacements (orthogonal directions to beamdirection).

In the preferred embodiment of the present invention, above-describedpurposes are achieved.

All the displacement measurement methods related to the presentinvention allow measuring local displacement vector or localdisplacement vector components from the phases of the ultrasound echosignals acquired from the target as the responses to more than one timetransmitted ultrasound.

One method measures the displacement vector component from the gradientof the cross-spectrum phase evaluated from echo signals acquired at twodifferent time, i.e., before and after tissue deformation. The 3Dprocessing yields from 3D cross-spectrum phase θ(ωx,ωy,ωz) accuratemeasurements of 3D displacement vectors ((d=(dx, dy, dz)^(T)) in 3D ROI,and consequently, results in measurements of the more accuratedisplacement vector components compared with corresponding ones measuredby 2D processing (2D cross-spectrum phase: θ(ωx,ωy), or θ(ωy,ωz), orθ(ωx,ωz)) and 1D processing (1D cross-spectrum phase: θ(ωx), or θ(ωy),or θ(ωz)).

When measuring displacement from the gradient of the echo cross-spectrumphase, to result the more accurate measurement accuracy, the leastsquares method can be applied with utilization as the weight function ofthe squares of the cross-spectrum usually normalized by thecross-spectrum power, where, to stabilize the measurement, theregularization method can be applied, by which a priori information canbe incorporated, i.e., about within the ROI the magnitude of the unknowndisplacement vector, spatial continuity and differentiability of theunknown displacement vector distribution, etc. The regularizationparameter depends on time-space dimension of the ROI, direction of theunknown displacement component, position of the unknown displacementvector, etc. Otherwise, directional independent regularization utilizethe mechanical properties of tissue (e.g., incompressibility) andcompatibility conditions of displacement vector distribution anddisplacement component distribution.

The displacement measurement apparatus related to the present inventioncan be equipped with the following means: displacement (strain) sensor(transducer to transmit ultrasounds to the target, and detect echosignals generated in the target), relative position controller andrelative direction controller between the sensor and the target, meansof transmitting/receiving (transmitter of driving signals to the sensor,and receiver of the echo signals detected at the sensor), means of dataprocessing (controller of the driving signals of the means oftransmitting, and processor of the received echo signals of means ofreceiving), and means of data storing (storage of echo signals, measureddeformation data).

The means of data processing also measures the local displacement vectoror the local displacement vector components utilizing the stateddisplacement measurement methods from the phases of the ultrasound echosignals acquired from the target as the responses to more than one timetransmitted ultrasound.

The strain tensor measurement apparatus related to the first point ofview of the present invention can be equipped with the displacementmeasurement apparatus, and the means of data processing can yield straintensor components by spatial differential filtering with suitable cutoff frequency in spatial domain or frequency domain the measured 3D, or2D displacement vector components, or measured one directiondisplacement component in the 3D, 2D, or 1D ROI. The means of dataprocessing can also yield strain rate tensor components, accelerationvector components, or velocity vector components by time differentialfiltering with suitable cut off frequency in time domain or frequencydomain the measured time series of displacement components, or straincomponents.

The strain tensor measurement method related to the present inventionalso allow directly measuring the local strain tensor or the localstrain tensor components from the phases of the ultrasound echo signalsacquired from the target as the responses to more than one timetransmitted ultrasound.

The strain tensor measurement apparatus related to the second point ofview of the present invention can be equipped with the following means:displacement (strain) sensor (transducer to transmit ultrasounds to thetarget, and detect echo signals generated in the target), relativeposition controller and relative direction controller between the sensorand the target, means of transmitting/receiving (transmitter of drivingsignals to the sensor, and receiver of the echo signals detected at thesensor), means of data processing (controller of the driving signals ofthe means of transmitting, and processor of the received echo signals ofmeans of receiving), and means of data storing (storage of echo signals,measured deformation data).

The means of data processing also directly measures the local straintensor or the local strain tensor components utilizing the stated directstrain measurement methods from the phases of the ultrasound echosignals acquired from the target as the responses to more than one timetransmitted ultrasound.

The elasticity and visco-elasticity constants measurement apparatusrelated to the first point of view of the present invention can beequipped with the following means: means of data storing (storage of atleast one of strain tensor data, strain rate tensor data, accelerationvector data, elastic constants, visco elastic constants, or densitymeasured in the ROI set in the target), and means of calculating elasticand visco elastic constants (calculator of at least one of elasticconstants, visco elastic constants, or density of arbitrary point in theROI from at least one of the measured strain tensor data, strain ratetensor data, or acceleration vector data).

The means of calculating elastic and visco elastic constants numericallydetermines at least one of the elastic constants, visco elasticconstants, or density from the first order partial differentialequations relating at least one of the elastic constants, visco elasticconstants, or density to at least one of the strain tensor data, strainrate tensor data, acceleration vector data. Time delays or relaxationtimes can also be determined by ratio of the corresponding elasticconstant and visco elastic constant.

The elasticity and visco-elasticity constants measurement apparatusrelated to the second point of view of the present invention can beequipped with the following means: means of data storing (storage of atleast one of strain tensor data, strain rate tensor data, accelerationvector data, elastic constants, visco elastic constants, or densitymeasured in the ROI including lesions), means of calculating elastic andvisco elastic constants (calculator of at least one of elasticconstants, visco elastic constants, or density of arbitrary point in theROI from at least one of the measured strain tensor data, strain ratetensor data, or acceleration vector data), and means of output ofdegeneration information on parts including the lesions (output means ofdegeneration information based on calculated at least one of the elasticconstants, visco elastic constants, or density).

The means of calculating elastic and visco elastic constants numericallydetermines at least one of the elastic constants, visco elasticconstants, or the density from the first order partial differentialequations relating at least one of the elastic constants, visco elasticconstants, or density to at least one of the strain tensor data, strainrate tensor data, acceleration vector data.

The elasticity and visco-elasticity constants measurementapparatus-based treatment apparatus related to the present invention canbe equipped with the following means: treatment transducer arrayed withmore than one oscillator, means (circuit) of treatment transmitting(transmitter of driving signals to each oscillator of the treatmenttransducer array), diagnosis transducer arrayed with more than oneoscillator, means (circuit) of diagnosis transmitting (transmitter ofdriving signals to each oscillator of the diagnosis transducer array),means (circuit) of receiving (receiver of the echo signals detected atthe oscillators of the transducers and matcher of the echo signals basedon their phases), means of calculating elastic and visco elasticconstants (calculator of at least one of elastic constants, viscoelastic constants, or density from the matched echo signals), means ofoutput of degeneration information on parts including the lesions(output means of degeneration information based on calculated at leastone of the elastic constants, visco elastic constants, or density),controller of the means (circuit) of treatment transmitting, means(circuit) of diagnosis transmitting, means (circuit) of receiving, andmeans of calculating elastic and visco elastic constants, and the inputmeans of commands and conditions into the controller.

The controller is not only equipped with functions for controlling themeans (circuit) of diagnosis transmitting and means (circuit) ofreceiving based on the commands and the conditions, but also withfunctions for deforming the ROI in the target based on the commands andthe conditions, and for controlling the means (circuit) of treatmenttransmitting to control the treatment ultrasound beam transmitted fromthe treatment transducer based on the commands and the conditions.

The means of calculating elastic and visco elastic constants obtains thematched echo signals in the ROI based on the commands given from thecontroller, and calculates at least one of strain tensor data, strainrate tensor data, or acceleration vector data in the ROI, andsubsequently calculates from these deformation data at least one ofelastic constants, visco elastic constants, or density in the ROI. Here,controlled of treatment ultrasound beam may be beam focus position,treatment interval, ultrasound beam power, ultrasound beam strength,transmit term, beam shape (apodization), etc. The oscillators may serveboth as treatment ones and diagnosis ones.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 shows a schematic representation of a global frame ofdisplacement vector and strain tensor measurement apparatus, andelasticity and visco-elasticity constants measurement apparatus, relatedto one of conduct forms of the present invention;

FIG. 2 shows illustration of a displacement (strain) sensor applicableto the present invention;

FIG. 3 shows illustration of mechanical scan movements of thedisplacement (strain) sensor;

FIG. 4 shows illustration of beam steering, and spatial interporation ofmeasured two displacement vector component distributions;

FIG. 5 shows illustration of transmitted beams whose amplitudes are sinemodulated in scan directions;

FIG. 6 shows illustration of a basic (n=1) wave component and n-thharmonic wave components (n equals from 2 to N) of ultrasound echosignal;

FIG. 7 shows illustration of a local 3D space centered on a point(x,y,z) in 3D ROI in pre-deformation ultrasound echo signal space, andthe shifted one in post-deformation ultrasound echo signal space;

FIG. 8 shows illustration as the example of searching for local 3Dultrasound echo signal by phase matching in searching space set inpost-deformation ultrasound echo signal space. That is, thecorresponding local signal is searched for using pre-deformation localecho signal;

FIG. 9 shows illustration to make 3D displacement vector distributionhigh spatial resolution, i.e., to make local space small;

FIG. 10 shows flowchart of method of 3D displacement vector distributionin 3D space (method 1-1), of method of 2D displacement vectordistribution in 2D region (method 2-1), of method of one directiondisplacement component distribution in 1D region (method 3-1);

FIG. 11 shows flowchart of method of 3D displacement vector distributionin 3D space (method 1-2), of method of 2D displacement vectordistribution in 2D region (method 2-2), of method of one directiondisplacement component distribution in 1D region (method 3-2);

FIG. 12 shows flowchart of method of 3D displacement vector distributionin 3D space (method 1-3), of method of 2D displacement vectordistribution in 2D region (method 2-3), of method of one directiondisplacement component distribution in 1D region (method 3-3);

FIG. 13 shows flowchart of method of 3D displacement vector distributionin 3D space (method 1-4), of method of 2D displacement vectordistribution in 2D region (method 2-4), of method of one directiondisplacement component distribution in 1D region (method 3-4);

FIG. 14 shows flowchart of method of 3D displacement vector distributionin 3D space (method 1-5), of method of 2D displacement vectordistribution in 2D region (method 2-5), of method of one directiondisplacement component distribution in 1D region (method 3-5);

FIG. 15 shows illustration of a local 2D region centered on a point(x,y) in 2D ROI in pre-deformation ultrasound echo signal space, and theshifted one in post-deformation ultrasound echo signal space;

FIG. 16 shows illustration as the example of searching for local 2Dultrasound echo signal by phase matching in searching region set inpost-deformation ultrasound echo signal space. That is, thecorresponding local signal is searched for using pre-deformation localecho signal;

FIG. 17 shows illustration to make 2D displacement vector distributionhigh spatial resolution, i.e., to make local region small;

FIG. 18 shows illustration of a local 1D region centered on a point (x)in 1D ROI in pre-deformation ultrasound echo signal space, and theshifted one in post-deformation ultrasound echo signal space;

FIG. 19 shows illustration as the example of searching for local 1Dultrasound echo signal by phase matching in searching region set inpost-deformation ultrasound echo signal space. That is, thecorresponding local signal is searched for using pre-deformation localecho signal;

FIG. 20 shows illustration to make one direction displacement componentdistribution high spatial resolution, i.e., to make local region small;

FIG. 21 shows flowchart of method of 2D displacement vector distributionin 3D space (method 4-1), of method of one direction displacementcomponent distribution in 3D space (method 5-1), and of method of onedirection displacement component distribution in 2D region (method 6-1);

FIG. 22 shows flowchart of method of 2D displacement vector distributionin 3D space (method 4-2), of method of one direction displacementcomponent distribution in 3D space (method 5-2), and of method of onedirection displacement component distribution in 2D region (method 6-2);

FIG. 23 shows flowchart of method of 2D displacement vector distributionin 3D space (method 4-3), of method of one direction displacementcomponent distribution in 3D space (method 5-3), and of method of onedirection displacement component distribution in 2D region (method 6-3);

FIG. 24 shows flowchart of method of 2D displacement vector distributionin 3D space (method 4-4), of method of one direction displacementcomponent distribution in 3D space (method 5-4), and of method of onedirection displacement component distribution in 2D region (method 6-4);

FIG. 25 shows flowchart of method of 2D displacement vector distributionin 3D space (method 4-5), of method of one direction displacementcomponent distribution in 3D space (method 5-5), and of method of onedirection displacement component distribution in 2D region (method 6-5);

FIG. 26 shows flowchart of measurement procedure of elasticityconstants, and visco-elasticity constants utilizing the elasticity andvisco-elasticity constants measurement apparatus (FIG. 1);

FIG. 27 shows a schematic representation of a global frame of elasticityand visco-elasticity constants measurement apparatus-based treatmentapparatus related to one of conduct forms of the present invention; and

FIG. 28 shows flowchart of control procedure of the elasticity andvisco-elasticity constants measurement apparatus-based treatmentapparatus (FIG. 27).

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The following is explanation in detail of conduct forms of the presentinvention with referring to figures.

FIG. 1 shows a schematic representation of a global frame ofdisplacement vector and strain tensor measurement apparatus, andelasticity and visco-elasticity constants measurement apparatus, relatedto one of conduct forms of the present invention. This apparatusmeasures in 3D, 2D, or 1D ROI 7 set in measurement object 6 displacementvector component distributions, strain tensor component distributions,their time-space partial derivative distributions, etc. to obtain straintensor field, strain rate tensor field, acceleration vector etc., fromwhich this apparatus measures following constant distributions, i.e.,elastic constants such as shear modulus, Poisson's ratio, etc., viscoelastic constants such as visco shear modulus, visco Poisson's ratio,etc., delay times or relaxation times relating these elastic constantsand visco elastic constants, or density.

As shown in FIG. 1, displacement (strain) sensor 5 can be directlycontacted to object surface, or suitable medium can be put between thesensor and the object. On this conduct form, as the displacement(strain) sensor, ultrasound transducer is used. The transducer may have1D or 2D array of oscillators.

Distance between the object 6 and the displacement (strain) sensor 5 canbe mechanically controlled by position controller 4. Moreover, relativedistance between the object 6 and the displacement (strain) sensor 5 canbe mechanically controlled by position controller 4′. Ultrasoundtransmitter (ultrasound pulser) 5′ is equipped to drive the displacement(strain) sensor 5, and 5′ also serves as output controller, i.e.,receiver with amplifier of echo signals detected at the displacement(strain) sensor 5. Furthermore, mechanical source 8 can be equipped toactively apply static compression, vibration, etc., and mechanicalposition controller 4′, can be also equipped.

Output echo signals of output controller 5′ are stored at storage 2passing through measurement controller 3. The echo signals stored atstorage 2 are read out by data processor 1, and displacement vectorcomponent distributions (time series) or strain tensor componentdistributions (time series) are directly calculated and obtained ofarbitrary time in the ROI 7, and further calculated and obtained are astheir time-space partial derivatives, i.e., strain tensor componentdistributions (time series), strain rate tensor component distributions(time series), acceleration vector component distributions (timeseries), etc. That is, when displacement vector component distributionsare calculated of the ROI 7, strain tensor component distributions (timeseries) are obtained by implementing 3D, 2D, or 1D spatial differentialfilter to the obtained displacement vector component distributions (timeseries). The cut off frequencies of all the filters used in the presentinvention can be set different-values freely at each point at each timein each spatio-temporal direction as those of usual filters. Theacceleration vector component distributions (time series) are obtainedby implementing time differential filter twice to the measureddisplacement vector component distributions (time series). The strainrate tensor component distributions (time series) are obtained byimplementing spatial differential filter to the velocity vectorcomponent distributions (time series) obtained by implementing timedifferential filter to the displacement vector component distributions(time series), or by implementing time differential filter once to themeasured strain tensor component distributions (time series). Moreover,when strain tensor component distributions (time series) are directlycalculated of the ROI 7 and obtained, strain rate tensor componentdistributions (times series) are obtained by implementing timedifferential filter to the measured strain tensor componentdistributions (time series). Furthermore, this data processor 1calculates following constant distributions, i.e., elastic constantssuch as shear modulus, Poisson's ratio, etc., visco elastic constantssuch as visco shear modulus, visco Poisson's ratio, etc., delay times orrelaxation times relating these elastic constants and visco elasticconstants, or density from the measured distributions of strain tensorcomponents (time series), strain rate tensor components (time series),acceleration vector components (time series), etc. These calculatedresults are stored at the storage 2.

The measurement controller 3 controls the data processor 1, the positioncontroller 4 and 4″, and the transmitting/output controller 5′. Theposition controller 4′ is not utilized when the object 6 is spatiallyfixed. When displacement (strain) sensor 5 is electronic scan type,position controller 4 is not always utilized. That is, it may bepossible to measure without mechanical scanning. The displacement(strain) sensor 5 may be contacted on the object 6, or may not. That is,the displacement (strain) sensor 5 and the object 6 may be dipped in orimmersed in water tank, for instance, when monitoring the treatmenteffectiveness of High Intensity Focus Ultrasound (HIFU).

The position controller 4 mechanically controls the relative positionbetween the displacement (strain) sensor 5 and the object 6.Specifically, the position controller 4 realizes vertical, horizon,turn, and fan direction scan movements (FIG. 3). The output of thetransmitting/output controller 5′ is also stored at storage 2successively or with given time intervals. The data processor 1 controlsthe transmitting/output controller 5′, and acquires the echo's basicwave components, n-th harmonic wave components (n equals from 2 to N),or all the components in 3D, 2D, or 1D ROI 7, and implementsbelow-described data processing to yield displacement data, strain data,strain rate data, or acceleration data, and stores measured these datain the storage 2.

The transmitting/output controller 5′ and the data processor 1 obeys thecommands of measurement controller 3, and carry out synthetic apertureprocessing, e.g., transmitting fixed focusing process,multi-transmitting fixed focusing process, receiving dynamic focusingprocess, etc. Furthermore, the transmitting/output controller 5′ and thedata processor 1 carry out apodization process of ultrasound signals,i.e., weighting process on each ultrasound transmitted/received at eachoscillator to sharpen the synthesized ultrasound beam, and carry outbeam steering process to acquire the echo signals of 3D, 2D, or 1D ROI.

Next explanation is in detail about displacement and strain measurementapparatus related to conduct forms of the present invention.

On this conduct form, as the displacement (strain) sensor 5, thefollowing type ultrasound transducers can be utilized, i.e., 2D arraybeing mechanical scan possible, 2D array being electronic scan possible,1D array being mechanical scan possible, and 1D array being electronicscan possible.

On this conduct form, synthetic aperture can be performed. Also beamsteering can be performed. When beam steering is performed, measureddisplacement component distributions and strain tensor componentdistributions are spatially interporated, after which these measureddisplacement component distributions (time series) and strain tensorcomponent distributions (time series) are time-spatially differentiatedto yield strain tensor component distributions (time series), strainrate tensor component distributions (time series), acceleration vectorcomponent distributions (time series), and velocity vector componentdistributions (time series).

As measurement of the beam direction is considerably accurate comparedwith that of the orthogonal scan direction, to yield high accuracydisplacement vector measurement, mechanical scan and/or beam steeringcan be performed. That is, echo data frames are acquired by performingmechanical scan and/or beam steering such that ultrasound beams aretransmitted in three different directions when measuring 3D displacementvector, and in two different directions when measuring 2D displacementvector. From two echo data frames acquired by transmitting theultrasound beams in same direction, accurately the distribution ofdisplacement component in beam direction is measured, by which accurate3D or 2D displacement vector distribution can be obtained (e.g., FIG.4).

However, to obtain the final displacement vector distribution,displacement vector distributions on the different old discretecoordinates must be converted to ones on one new discrete coordinate.That is, by interporating the displacement component distributionmeasured on the old discrete coordinate, the displacement component canbe obtained at the point of the new discrete coordinate. Concretely,displacement component distribution is Fourier's transformed, which ismultiplied with complex exponential such that the phase is shifted.Thus, realized is spatial shifting of the displacement componentdistribution.

On this conduct form, amplitudes of transmitted beams can be sinemodulated in scan directions.

The sine modulation frequency is better to be higher. However, as thismodulation shifts in frequency domain in scan direction the banddetermined by beam width, based on the sampling theorem the modulationfrequency needs to be set such that the highest frequency becomes lessthan the half of the sampling frequency determined by beam pitch. Thus,improved is measurement accuracy of displacement component distributionin scan direction being orthogonal to beam direction.

Based on these processes, obtained ultrasound echo signals in 3D, 2D, or1D ROI can be effectively utilized, i.e., basic wave components,harmonic wave components (The carrier frequency higher, improved ismeasurement accuracy of displacement component in beam direction. Thecarrier frequency higher, the beam width narrower. Thus, as thebandwidth in scan direction is wider compared with the basic componentwave, also improved is measurement accuracy of displacement component inscan direction.), or all the wave components due to low SNRs of onlyharmonic wave components.

That is, below-described displacement and strain measurement methods canutilize the ultrasound echo signals, or only extracted the basic wavecomponents, or only extracted the n-th harmonic wave components (nequals from 2 to N), or these combinations (methods from 1-1 to 1-5,from 2-1 to 2-5, from 3-1 to 3-5, from 4-1 to 4-5, from 5-1 to 5-5, from6-1 to 6-5.).

These stated displacement and strain measurement methods base oniteratively updating the displacement estimate utilizing the estimatedremaining error data (estimated residual displacement data). The initialestimate is set based on the a priori knowledge about measurementtarget, i.e., displacement distribution, strain distribution, strainrate distribution, acceleration distribution, or velocity distribution.Finally obtained are accurate displacement vector distribution (timeseries), displacement vector component distributions (time series),strain tensor distribution (time series), strain tensor componentdistributions (time series), strain rate tensor distribution (timeseries), strain rate tensor component distributions (time series),acceleration vector distribution (time series), acceleration vectorcomponent distributions (time series), velocity vector distribution(time series), or velocity vector component distributions (time series).

However, when stressing on real-time processing, measurement can befinished only with the once estimation.

During iterative estimation of the displacement vector and residualdisplacement vector, when estimation errors are detected a priori as thepoints of time-space magnitude and time-space continuity, for instance,the estimates can be cut by compulsion such that the estimates rangefrom the given smallest value to the given largest value, or such thatthe difference between the estimates of the neighboring points settlewithin the given ranges.

On these stated iterative displacement and strain measurement methods,all the methods for estimating the residual displacement vectorcomponent or the displacement vector component utilize as the index thephases of the ultrasound echo signals acquired at more than one time.First of all, one of these methods is used to explain the iterativemethods, i.e., the method estimating displacement from the gradient ofthe phase of the cross-spectrum of ultrasound echo signals acquiredtwice.

The displacement and strain measurement methods can be implemented eachon extracted the basic wave signals and the n-th harmonic wavecomponents (n equals from 2 to N). In this case, the final measurementresult can be obtained as the mean displacement data weighted by thepower ratio of the cross-spectrums etc.

In addition, when measuring the displacement from the gradient of thecross-spectrum phase utilizing least squares method, data processor alsoutilize the regularization method based on the a priori knowledge, whichimproves stability, accuracy, and spatial resolutions of the measurementof the displacement vector distribution, or the displacement vectorcomponent distributions.

In the past, when large displacement needs to be handled, beforeestimating the gradient of the cross-spectrum phase, the phase had beenunwrapped, or the displacement had been coarsely estimated bycross-correlation method. Thus, measurement procedure had become complexone. To cope with these complexity, the measurement procedure is madesimpler without these processes by introducing process of thinning outdata and process of remaking data interval original. Thus, implementedsoft amount and calculation time are reduced. Occasionally, theregularization may not be performed.

However, as other method, before estimating the gradient of thecross-spectrum phase, the phase can also be unwrapped, or thedisplacement can also be coarsely estimated by cross-correlation method.Also in this case, when measuring the local displacement from thegradient of the cross-spectrum phase, a priori knowledge about thedisplacement distribution in the ROI can be incorporated by utilizingthe regularization method, where the least squares method utilizes asthe weight function the squares of the cross-spectrum usually normalizedby the cross-spectrum power. Freely, when estimating the gradient of thecross-spectrum phase, acquired ultrasound echo signals can be thinnedout in each direction with constant intervals.

These cases handles the gradient of the local 3D, 2D or 1Dcross-spectrum phase evaluated on 3D, 2D, or 1D ultrasound echo signalsacquired at more than one time from 3D space, 2D or 1D region in theobject. Stably measured with high accuracy and high spatial resolutionsare 3D displacement vector component distributions in the 3D SOI (spaceof interest), 2D displacement vector component distributions in the 2DROI, one direction displacement component distribution in the 1D ROI, 2Ddisplacement vector component distributions or one directiondisplacement component distribution in the 3D SOI, or one directiondisplacement component distribution in the 2D ROI.

The displacement and strain measurement apparatus of the presentinvention measures in the 3D SOI, 2D, or 1D ROI in the object thedisplacement vector distribution, the strain tensor distribution, thestrain rate tensor distribution, the acceleration vector distribution,velocity vector distribution, etc. from ultrasound echo signals measuredin 3D SOI, 2D, or 1D ROI (referred to 3D, 2D, 1D ultrasound echosignals). The displacement and strain measurement apparatus can beequipped with:

displacement (strain) sensor (ultrasound transducer),

relative position controller and relative direction controller betweenthe sensor and the target (vertical, horizon, turn, and fan directionscan movements),

transmitter (ultrasound pulser)/output controller (receiver andamplifier),

means of data processing (synthetic aperture process: transmitting fixedfocusing process, multi-transmitting fixed focusing process, receivingdynamic focusing process etc., apodization),

means of data storing (storage of echo signals),

means of (signal) data processing (calculation of displacement vectordistribution, strain tensor distribution, strain rate tensordistribution, acceleration vector distribution, velocity vectordistribution, etc.), and

means of data storing (storage of the displacement vector distribution,strain tensor distribution, strain rate tensor distribution,acceleration vector distribution, velocity vector distribution, etc.).

In this case, the means of data processing can yield strain tensorcomponents by implementing spatial 3D, 2D, or 1D differential filterwith cut off frequency or multiplying Fourier's transform of thedifferential filter in frequency domain to 3D displacement vectorcomponent distributions in the 3D SOI (space of interest), 2Ddisplacement vector component distributions in the 2D ROI, one directiondisplacement component distribution in the 1D ROI, 2D displacementvector component distributions or one direction displacement componentdistribution in the 3D SOI, or one direction displacement componentdistribution in the 2D ROI. Moreover, by implementing time differentialfilter with cut off frequency or multiplying Fourier's transform of thedifferential filter in frequency domain to time series of these, thestrain rate tensor component distributions, acceleration vectorcomponent distributions, velocity vector component distributions.Moreover, the strain rate tensor component distributions can be obtainedfrom directly measured strain tensor component distributions.

The displacement and strain measurement apparatus can be also equippedwith static compressor or vibrator as mechanical source to generate atleast one strain tensor field (one displacement vector field) in the 3DSOI, 2D, or 1D ROI in the object. On this case, generated due to bodymotion (heart motion, blood vessel motion, respiratory), the straintensor field (displacement vector field) can be also measured in the 3DSOI, 2D, or 1D ROI in the object.

The following ultrasound transducer type can be utilized, i.e.,ultrasound oscillator being mechanical scan possible, electronic scantype 2D ultrasound oscillator array (being mechanical scan possible),and 1D ultrasound oscillator array (being mechanical scan possible).Thus, echo signal is synthesized one. When the displacement (strain)sensor is contacted on the object, the contact part can becomemechanical source. That is, the displacement (strain) sensor also servesas compressor or vibrator. When the part of lesion is dipped in orimmersed in water tank to carry out treatment with High Intensity FocusUltrasound (HIFU), the object can be non-contactly measured by dippingin or immersing the displacement (strain) sensor as well in water tank.

Moreover, when the displacement (strain) sensor is directly contacted toobject surface as mechanical source to stably measure elastic constantdistributions and visco elastic constant distributions, suitablereference medium can be put between the sensor and the object. In thiscase, the reference medium can also be mounted (installed) on thetransducer.

Basically, the means of data processing can yield strain tensorcomponent distributions, stain rate tensor component distributions,acceleration vector component distributions, or velocity vectorcomponent distributions from the obtained deformation data utilizing thedisplacement (strain) sensor from synthesized ultrasound echo in 3D SOI,2D or 1D ROI, i.e., 3D displacement vector component distributions inthe 3D SOI, 2D displacement vector component distributions in the 2DROI, one direction displacement component distribution in the 1D ROI, 2Ddisplacement vector component distributions or one directiondisplacement component distribution in the 3D SOI, or one directiondisplacement component distribution in the 2D ROI. Moreover, the strainrate tensor component distributions can be obtained from directlymeasured strain tensor component distributions.

In this case, the means of data processing can yield displacementcomponent distributions and strain tensor component distributions fromultrasound echo signals acquired in each dimensional ROI with beamsteering as well as synthetic aperture processing, from which obtainedcan be strain tensor component distributions, strain rate tensorcomponent distributions, acceleration vector component distributions,and velocity vector component distributions.

Moreover, in this case, the means of data processing can yielddisplacement component distributions and strain tensor componentdistributions from ultrasound echo basic wave components, ultrasoundecho harmonic wave components, or all the ultrasound echo components,from which obtained can be strain tensor component distributions, strainrate tensor component distributions, acceleration vector componentdistributions, and velocity vector component distributions.

Here, the sine modulation frequency is better to be higher. However, asthis modulation shifts in scan direction in frequency domain the banddetermined by beam width, based on the sampling theorem the modulationfrequency needs to be set such that the highest frequency becomes lessthan the half of the sampling frequency determined by beam pitch.

Furthermore, ultrasound echo signals can be acquired by combining theprocessing, i.e., synthetic aperture processing, beam steering, sinemodulation of transmitted beams' amplitudes in scan directions. In thiscase, measured can be displacement vector component distribution fromultrasound echo basic wave components, ultrasound echo harmonic wavecomponents, or all the ultrasound echo components.

When utilizing below-described displacement and strain measurementmethods, as measurement of the beam direction is considerably accuratecompared with that of the orthogonal scan direction, to yield highaccuracy displacement measurements, mechanical scan and/or beam steeringare performed. That is, echo data frames are acquired under object'spre- and post-deformation by performing mechanical scan and/or beamsteering such that ultrasound beams are transmitted in three differentdirections when measuring 3D displacement vector, and in two differentdirections when measuring 2D displacement vector. From two echo dataframes acquired by transmitting the ultrasound beams in same direction,accurately the distribution of displacement component in beam directionis measured, by which accurate 3D or 2D displacement vector distributionis obtained. To obtain the final displacement vector distribution,displacement vector distributions on the different old discretecoordinates must be converted to ones on one new discrete coordinate.That is, by interporating the displacement component distributionmeasured on the old discrete coordinate, the displacement component canbe obtained at the point of the new discrete coordinate. Concretely,displacement component distribution is Fourier's transformed, which ismultiplied with complex exponential such that the phase is shifted.Thus, realized is spatial shifting of the displacement componentdistribution. Strain tensor component distributions can be obtained fromthese displacement measurement data. Moreover, from these time series,obtained can be strain tensor rate component distributions, accelerationvector component distributions, velocity vector component distributions.Other displacement measurement methods and strain measurement methodscan be also applied to the ultrasound echo time series data in similarways.

Next explanation is in detail about displacement and strain measurementalgorithm related to conduct forms of the present invention. The meansof data processing 1 always carries out the below-explained calculationprocess or their combination, or as occasion demands.

-   (1) Calculation process of 3D displacement vector component    distribution in 3D ROI (below-described methods from 1-1 to 1-5)-   (2) Calculation process of 2D displacement vector component    distribution in 2D ROI (below-described methods from 2-1 to 2-5)-   (3) Calculation process of 1D (one direction) displacement component    distribution in 1D ROI (below-described methods from 3-1 to 3-5)-   (4) Calculation process of 2D displacement vector component    distribution in 3D ROI (below-described methods from 4-1 to 4-5)-   (5) Calculation process of 1D (one direction) displacement component    distribution in 3D ROI (below-described methods from 5-1 to 5-5)-   (6) Calculation process of 1D (one direction) displacement component    distribution in 2D ROI (below-described methods from 6-1 to 6-5)

When beam steering is performed, at means data processing 1, measureddisplacement vector component distributions are spatially interporated.

With respect to displacement component distributions and straincomponent distributions obtained through the above calculationprocesses, means of data processing 1 performs differentiation such thatthe followings are obtained, i.e., at each time strain tensor componentdistributions, strain gradient component distributions, strain ratetensor component distributions, strain rate gradient component,acceleration vector component distributions, or velocity vectordistributions. These calculated results are stored at storage 2.Moreover, these calculated results are displayed on display apparatussuch as CRT (color or gray scaled) in real-time or in quasi real-time.

As static or motion image, or time course (difference) image, etc., thefollowings can be displayed, i.e., displacement vector distribution,displacement vector component distributions, strain tensor componentdistributions, strain gradient component distributions, strain ratetensor component distributions, strain rate gradient component,acceleration vector component distributions, or velocity vectorcomponent distributions. At arbitrary points the values and their graph(time course) can also be displayed. For instance, by utilizingultrasound diagnosis apparatus, spatial variations of bulk modulus anddensity of tissues can be displayed in real-time. Thus, theabove-described static or motion image, or time course image of thedisplacement vector distribution, etc. can also be superimposed anddisplayed on the ultrasound image. The followings can be displayed invector style as well, i.e., the displacement vector distribution,acceleration vector, velocity vector.

The following is explanation in detail of displacement measurement andcalculation processes.

(I) Method 1: Measurement of 3D Displacement Vector Distribution

3D displacement vector distribution can be measured in 3D SOI 7 in theCartesian coordinate system. 3D ultrasound echo signals are acquiredunder pre-deformation and post-deformation. These echo signals areprocessed by the below-described methods 1-1, 1-2, 1-3, 1-4, and 1-5.That is, as shown in FIG. 7, local space is set at each point in thepre- and post-deformation 3D echo signal, and as shown in FIG. 8, thecorresponding local space is iteratively searched for in the SOI 7 usingthe local phase characteristics as the index. In this searching scheme,the estimated residual displacement vector is used to update thepreviously estimated displacement vector. When the estimated residualdisplacement vector satisfies with prescribed condition, the local spacesize is made small (FIG. 9). Thus, accurate 3D displacement vectormeasurement is realized. Here, sampling intervals are Δx, Δy, Δzrespectively in the x, y, and z-axes.

[Method 1-1]

The procedure of the method 1-1 is shown in FIG. 10. The processes from1 to 5 yields 3D displacement vector d(x,y,z)[=(dx(x,y,z), dy(x,y,z),dz(x,y,z))^(T)] of arbitrary point (x,y,z) in 3D SOI from pre- andpost-deformation local 3D echo signals r₁(l,m,n) and r₂(l,m,n) [0≦l≦L−1,0≦m≦M−1, 0≦n≦N−1] centered on (x,y,z) of pre- and post-deformation 3Decho signals r₁(x,y,z) and r₂(x,y,z). L, M, and N should be determinedsuch that ΔxL, ΔyM, ΔzN are respectively at least 4 times longer thancorresponding displacement components |dx(x,y,z)|, |dy(x,y,z)|,|dz(x,y,z)|.

(Process 1: Phase Matching at the Point (x,y,z))

Phase matching is performed to obtain i-th estimated^(i)(x,y,z)[=(d^(i)x(x,y,z), d^(i)y(x,y,z), d^(i)z(x,y,z))^(T)] of the3D displacement vector d(x,y,z)[=(dx(x,y,z), dy(x,y,z), dz(x,y,z))^(T)].

Searching space is set in the post-deformation echo signal spacer₂(x,y,z), being centered on the local space [0≦l≦L−1, 0≦m≦M−1, 0≦n≦N−1]centered on (x,y,z) and being twice longer than the correspondinglength, in order to update the i−1 th estimated^(i−1)(x,y,z)[=(d^(i−1)x(x,y,z), d^(i−1)y(x,y,z), d^(i−1)z(x,y,z))^(T)]of the 3D displacement vector d(x,y,z)[=(dx(x,y,z), dy(x,y,z),dz(x,y,z))^(T)], whered ⁰(x,y,z)={hacek over (d)}(x,y,z).  (1)

The phase of the post-deformation local echo signal is matched topre-deformation local echo signal by multiplying

$\begin{matrix}{\exp\left\{ {{j\frac{2\pi}{L}\frac{d_{x}^{i - 1}\left( {x,y,z} \right)}{\Delta\; x}1} + {j\frac{2\pi}{M}\frac{d_{y}^{i - 1}\left( {x,y,z} \right)}{\Delta\; y}m} + {j\frac{2\pi}{N}\frac{d_{z}^{i - 1}\left( {x,y,z} \right)}{\Delta\; z}n}} \right\}} & (2)\end{matrix}$to 3D Fourier's transform of this searching space echo signal r′₂(l,m,n)[0≦l≦2L−1, 0≦m≦2M−1, 0≦n≦2N−1] using i-th estimate d^(i−1)(x,y,z), or bymultiplying

$\begin{matrix}{\exp\left\{ {{j\frac{2\pi}{L}\frac{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}{\Delta\; x}l} + {j\frac{2\pi}{M}\frac{{\overset{\Cap}{u}}_{y}^{i - 1}\left( {x,y,z} \right)}{\Delta\; y}m} + {j\frac{2\pi}{N}\frac{{\overset{\Cap}{u}}_{z}^{i - 1}\left( {x,y,z} \right)}{\Delta\; z}n}} \right\}} & \left( 2^{\prime} \right)\end{matrix}$to 3D Fourier's transform of the i−1 th phase-matched searching spaceecho signal r′^(i−1) ₂(l,m,n) using the estimate

^(i−1)(x,y,z)[=(

_(x) ^(i−1)(x,y,z),

_(y) ^(i−1)(x,y,z),

_(z) ^(i−1)(x,y,z))^(T)] [

⁰(x,y,z)=0 (zero vector)] of the vector u^(i−1)(x,y,z)[=(u^(i−1)_(x)(x,y,z), u^(i−1) _(y)(x,y,z), u^(i−1) _(z)(x,y,z))^(T)].

By carrying out inverse Fourier's transform of this product,post-deformation echo signal r^(i) ₂(l,m,n) is obtained at the center ofthe searching space echo signal r′^(i) ₂(l,m,n) which is used at i-thstage to estimate 3D displacement vector d(x,y,z)[=(dx(x,y,z),dy(x,y,z), dz(x,y,z))^(T)].

Alternatively, the phase of the pre-deformation local echo signal can bematched to post-deformation local echo signal in a similar way. That is,3D Fourier's transform of the searching space echo signal r′₁(l,m,n)[0≦l≦2L−1, 0≦m≦2M−1, 0≦n≦2N−1] centered on the point (x,y,z) in thepre-deformation echo signal space is multiplied with

$\begin{matrix}{{\exp\left\{ {{{- j}\frac{2\pi}{L}\frac{d_{x}^{i - 1}\left( {x,y,z} \right)}{\Delta\; x}1} - {j\frac{2\pi}{M}\frac{d_{y}^{i - 1}\left( {x,y,z} \right)}{\Delta\; y}m} - {j\frac{2\pi}{N}\frac{d_{z}^{i - 1}\left( {x,y,z} \right)}{\Delta\; z}n}} \right\}},} & \left( 2^{\prime\prime} \right)\end{matrix}$or 3D Fourier's transform of the i−1 th phase-matched searching spaceecho signal r′^(i−1) ₁(l,m,n) is multiplied with

$\begin{matrix}{\exp{\left\{ {{{- j}\frac{2\pi}{L}\frac{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}{\Delta\; x}l} - {j\frac{2\pi}{M}\frac{{\overset{\Cap}{u}}_{y}^{i - 1}\left( {x,y,z} \right)}{\Delta\; y}m} - {j\frac{2\pi}{N}\frac{{\overset{\Cap}{u}}_{z}^{i - 1}\left( {x,y,z} \right)}{\Delta\; z}n}} \right\}.}} & \left( 2^{\prime\prime\prime} \right)\end{matrix}$(Process 2: Estimation of 3D Residual Displacement Vector at the Point(x,y,z))

Local 3D echo cross-spectrum is evaluated from the 3D Fourier'stransforms of the pre-deformation local 3D ultrasound echo signalr₁(l,m,n) and phase-matched post-deformation local 3D ultrasound echosignal r^(i) ₂(l,m,n)S ^(i) _(2,1)(l,m,n)=R ₂ ^(i)*(l,m,n)R ₁(l,m,n),  (3)where * denotes conjugate.

Alternatively, when pre-deformation local 3D ultrasound echo signal isphase-matched, cross-spectrum of r^(i) ₁(l,m,n) and r₂(l,m,n) isevaluated asS ^(i) _(2,1)(l,m,n)=R ₂*(l,m,n)R ^(i) ₁(l,m,n).Cross-spectrum is represented as

$\begin{matrix}{{{S_{2,1}^{i}\left( {l,m,n} \right)} \cong {{{R_{1}^{\prime}\left( {l,m,n} \right)}}^{2}\exp\left\{ {{j\frac{2\pi}{L}\frac{u_{x}^{i}\left( {x,y,z} \right)}{\Delta\; x}l} + {j\frac{2\pi}{M}\frac{u_{y}^{i}\left( {x,y,z} \right)}{\Delta\; y}m} + {j\frac{2\pi}{M}\frac{u_{z}^{i}\left( {x,y,z} \right)}{\Delta\; z}n}} \right\}}},} & (4)\end{matrix}$where 0≦l≦L−1, 0≦m≦M−1, 0≦n≦N−1,and then the phase is represented as

$\begin{matrix}{{{\theta^{i}\left( {l,m,n} \right)} = {\tan^{- 1}\left( \frac{{Im}\left\lbrack {S_{2,1}^{i}\left( {l,m,n} \right)} \right\rbrack}{{Re}\left\lbrack {S_{2,1}^{i}\left( {l,m,n} \right)} \right\rbrack} \right)}},} & (5)\end{matrix}$where Re[•] and Im[•] respectively represent the real and imaginarycomponent of “•”.

The least squares method is implemented on the gradient of the phase eq.(5) weighted with squared cross-spectrum |S_(2,1) ^(i)(l,m,n)|²(=Re²[S_(2,1) ^(i) (l,m,n)]²+Im²[S_(2,1) ^(i) (l,m,n)]). Thatis, by minimizing functional

error (u^(i)(x,y,z))

$\begin{matrix}{= {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2} \times \begin{matrix}\left( {\theta^{\prime}\left( {l,{\left( {m,n} \right) - {{u_{x}^{i}\left( {x,y,z} \right)}\left( \frac{2\pi}{L\;\Delta\; x} \right)l} -^{\mspace{110mu}}}} \right.} \right. \\\left. {{{u_{y}^{i}\left( {x,y,z} \right)}\left( \frac{2\pi}{M\;\Delta\; y} \right)m} - {{u_{z}^{i}\left( {x,y,z} \right)}\left( \frac{2\pi}{N\;\Delta\; z} \right)n}} \right)^{2}\end{matrix}}}} & (6)\end{matrix}$with respect to the 3D residual vector u^(i)(x,y,z) to be used to updatethe i−1 th estimate d^(i−1)(x,y,z) of the 3D displacement vectord(x,y,z), the estimate of u^(i)(x,y,z) is obtained as

^(i)(x,y,z)[=(

_(x) ^(i)(x,y,z),

_(y) ^(i)(x,y,z),

_(z) ^(i)(x,y,z))^(T)].  (6.2)

Concretely, the next simultaneous equations are solved.

$\begin{matrix}{\begin{bmatrix}{\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)l\;{\theta^{\prime}\left( {l,m,n} \right)}}} \\{\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{M\;\Delta\; y} \right)m\;{\theta^{\prime}\left( {l,m,n} \right)}}} \\{\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{N\;{\Delta z}} \right)n\;{\theta^{\prime}\left( {l,m,n} \right)}}}\end{bmatrix} = {\begin{bmatrix}{\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)^{2}l^{2}}} & {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)\left( \frac{2\pi}{M\;\Delta\; y} \right){lm}}} & {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)\left( \frac{2\pi}{N\;\Delta\; z} \right)\ln}} \\{\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)\left( \frac{2\pi}{M\;\Delta\; y} \right){lm}}} & {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{M\;\Delta\; x} \right)^{2}m^{2}}} & {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{M\;\Delta\; y} \right)\left( \frac{2\pi}{N\;\Delta\; z} \right){mn}}} \\{\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)\left( \frac{2\pi}{N\;\Delta\; z} \right)\ln}} & {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{M\;\Delta\; y} \right)\left( \frac{2\pi}{N\;\Delta\; z} \right){mn}}} & {\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}\left( \frac{2\pi}{N\;\Delta\; z} \right)^{2}n^{2}}}\end{bmatrix} \times \begin{bmatrix}{u_{\; x}^{\; i}\left( {x,y,z} \right)} \\{u_{y}^{i}\left( {x,y,z} \right)} \\{u_{z}^{i}\left( {x,y,z} \right)}\end{bmatrix}}} & (7)\end{matrix}$

When the 3D displacement vector d(x,y,z) is large, the 3D residualdisplacement vector u^(i)(x,y,z) needs to be estimated after unwrappingthe phase of the cross-spectrum [eq. (3)] in the frequency domain (l, m,n).

Alternatively, when the 3D displacement vector d(x,y,z) is large, byusing cross-correlation method (evaluation of the peak position of thecross-correlation function obtained as 3D inverse Fourier's transform ofthe cross-spectrum [eq. (3)]) at the initial stages during iterativeestimation, the 3D residual displacement vector u^(i)(x,y,z) can beestimated without unwrapping the phase of the cross-spectrum [eq. (3)]in the frequency domain. Specifically, by using the cross-correlationmethod, x, y, and z components of the 3D displacement vector arerespectively estimated as integer multiplications of the ultrasound echosampling intervals Δx, Δy, Δz. For instance, with respect to thresholdvalues correTratio or correTdiff, after

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {correTratio}}{or}} & (8) \\{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}} \leq {correTdiff}} & \left( 8^{\prime} \right)\end{matrix}$is satisfied with where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimate of the residual vectors, by using the estimate of the 3Ddisplacement vector d(x,y,z) as the initial estimate, the 3D residualdisplacement vector is estimated from the gradient of the phase of thecross-spectrum [eq. (3)].

Empirically it is known that after using cross-correlation method theconditions |u^(i) _(x)(x,y,z)|≦Δx/2, |u^(i) _(y)(x,y,z)|≦Δy/2, |u^(i)_(z)(x,y,z)|≦Δz/2 are satisfied with. However, for allowing estimationof the 3D residual displacement vector without unwrapping the phase ofthe cross-spectrum,

the necessary and sufficient condition is

$\begin{matrix}{{{{\frac{u_{x}^{i}\left( {x,y,z} \right)}{\Delta\; x} + \frac{u_{y}^{i}\left( {x,y,z} \right)}{\Delta\; y} + \frac{u_{z}^{i}\left( {x,y,z} \right)}{\Delta\; z}}} \leq 1}{or}} & (9) \\{{{{u_{x}^{i}\left( {x,y,z} \right)}} \leq {\Delta\;{x/3}}},{{{u_{y}^{i}\left( {x,y,z} \right)}} \leq {\Delta\;{y/3}}},{{{and}\mspace{11mu}{{u_{z}^{i}\left( {x,y,z} \right)}}} \leq {\Delta\;{z/3.}}}} & \left( 9^{\prime} \right)\end{matrix}$

Therefore, when estimating the gradient of the cross-spectrum phaseafter using cross-correlation method, the acquired ultrasound echo dataare thinned out with constant interval in each direction and the reducedecho data are used such that the condition (9) or (9′) is satisfiedwith. The iteration number i increasing, i.e., the magnitude of the 3Dresidual displacement vector components u^(i) _(x)(x,y,z), u^(i)_(y)(x,y,z), u^(i) _(z)(x,y,z) decreasing, the ultrasound echo datadensities are made restored in each direction. Hence, at initial stageswhere estimating the gradient of the cross-spectrum phase, for instance,ultrasound echo signals are used with one and half times or twice as along interval as the original interval in each direction. The densitiesof the ultrasound echo signals are made restored in each direction, forinstance, one and half times or twice per iteration.

Alternatively, when the magnitude of the 3D displacement vector d(x,y,z)is large, at initial stages, the acquired original ultrasound echo datacan be thinned out with constant interval in each direction and thereduced echo data can be used such that the 3D residual displacementvector can be estimated without unwrapping the phase of thecross-spectrum [eq. (3)] in the frequency domain (l,m,n). Specifically,the acquired original ultrasound echo data are thinned out with constantinterval in each direction and the reduced ehco data are used such thatthe condition (9) or (9′) is satisfied with. The iteration number iincreasing, i.e., the magnitude of the 3D residual displacement vectorcomponents u^(i) _(x)(x,y,z), u^(i) _(y)(x,y,z), u^(i) _(z)(x,y,z)decreasing, the ultrasound echo data densities are made restored in eachdirection, for instance, twice per iteration. When the 3D residualdisplacement vector components u^(i) _(x)(x,y,z), u^(i) _(y)(x,y,z),u^(i) _(z)(x,y,z) are estimated, if neither the condition (9) nor (9′)is satisfied with, the values are truncated such that the conditions aresatisfied with.

The interval of the ultrasound echo signal data are shortened, forinstance, when with respect to threshold values stepTratio or stepTdiffthe condition

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {stepTratio}}{or}} & (10) \\{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}} \leq {stepTdiff}} & \left( 10^{\prime} \right)\end{matrix}$is satisfied with, where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (10) or (10′) can be applied to each direction component,and in this case the data interval is shorten in each direction. Theseare also applied to below-described methods 1-2, 1-3, 1-4, and 1-5.

(Process 3: Update of the 3D Displacement Vector Estimate of the Point(x,y,z))

Thus, the i th estimate of the 3D displacement vector d(x,y,z) isevaluated asd ^(i)(x,y,z)=d ^(i−1)(x,y,z)+

^(i)(x,y,z)  (11)[Process 4: Condition for Heightening the Spatial Resolution of the 3DDisplacement Vector Distribution Measurement (Condition for Making theLocal Space Small)]

In order to make the spatial resolution high of the 3D displacementvector distribution measurement, the local space is made small duringiterative estimation. The criteria is below-described. The processes 1,2 and 3 are iteratively carried out till the criteria is satisfied with.When the criteria is satisfied with, the local space is made small, forinstance, the length of each side is made half. For instance, thecriteria is (12) or (12′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (12) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 12^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (12) or (12′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 5: Condition for Terminating the Iterative Estimation of the 3DDisplacement Vector of the Point (x,y,z))

Below-described is the criteria for terminating the iterative estimationof the 3D displacement vector of each point. The processes 1, 2 and 3are iteratively carried out till the criteria is satisfied with. Forinstance, the criteria is (13) or (13′) with respect to threshold valuesaboveTratio or aboveTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {aboveTratio}}{or}} & (13) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i\; - \; 1}\left( {x,y,z} \right)}}} \leq {{above}\;{Tdiff}}},} & \left( 13^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.(Process 6)

The 3D displacement vector component distributions are obtained bycarrying out processes 1, 2, 3, 4, and 5 at every point in the 3D SOI.

The initial estimate [eq. (1)] of the iterative estimation of the 3Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Limitation of Method 1-1]

The estimate of the 3D displacement vector d(x,y,z) is iterativelyupdated at each point (x,y,z) in the 3D SOI. Being dependent on the SNRof the local 3D echo signals, particularly at initial stages errorspossibly occur when estimating the residual vector and then phasematching possibly diverges. For instance, when solving eq. (7) [process2] or detecting the peak position of the cross-correlation function[process 2], errors possibly occur.

The possibility for divergence of the phase matching is, for instance,confirmed by the condition (14) or (14′) with respect to the thresholdvalue belowTratio or BelowTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {belowTratio}}{or}} & (14) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i\; - \; 1}\left( {x,y,z} \right)}}} \leq {{below}\;{Tdiff}}},} & \left( 14^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

To prevent phase matching (process 1) from diverging, in thebelow-described methods 1-2, 1-3, 1-4, and 1-5, by freely using thecondition (14) or (14′), estimation error is reduced of the residualvector. Thus, even if the SNR of the ultrasound echo signals are low,accurate 3D displacement vector measurement can be realized.

[Method 1-2]

The flowchart of the method 1-2 is shown in FIG. 11. To prevent phasematching from diverging at the process 1 of the method 1-1, estimationerror is reduced of the residual vector. Thus, even if the SNR of theultrasound echo signals are low, accurate 3D displacement vectormeasurement can be realized.

The procedure of iterative estimation is different from that of themethod 1-1. At i th estimate (i≧1), the following processes areperformed.

(Process 1: Estimation of the 3D Residual Displacement VectorDistribution)

Phase matching and estimation of the 3D residual displacement vector areperformed at every point (x,y,z) in the 3D SOI. That is, the processes 1and 2 of the method 1-1 are performed once at every point in the SOI.Thus, the estimate of the 3D residual vector distribution is obtained[eq. (6-2)].

(Process 2: Update of the Estimate of the 3D Displacement VectorDistribution)

The i−1 th estimate of the 3D displacement vector distribution isupdated using i th estimate of the 3D residual vector distribution.d ^(i)(x,y,z)=

^(i−1)(x,y,z)+

^(i)(x,y,z)  (15)

Next, this estimate is 3D low pass filtered or 3D median filter to yieldthe estimate of the 3D displacement vector distribution:

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED[d ^(i)(x,y,z)].  (16)

Thus, the estimation error is reduced of the residual vector comparedwith process 2 of the method 1-1 [eq. (7)]. Hence, phase matching of theprocess 1 of method 1-2 is performed using smoothed estimate of the 3Ddisplacement vector distribution.

[Process 3: Condition for Heightening the Spatial Resolution of the 3DDisplacement Vector Distribution Measurement (Condition for Making theLocal Space Small)]

In order to make the spatial resolution high of the 3D displacementvector distribution measurement, during iterative estimation, the localspace used for each point is made small, or the local space used overthe SOI is made small.

The criteria for each point is below-described. The processes 1 and 2(method 1-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local space is madesmall, for instance, the length of each side is made half. For instance,the criteria is (17) or (17′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (17) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 17^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (17) or (17′) can be applied to each direction component,and in this case the side is shorten in each direction.

The criteria over the SOI is below-described. The processes 1 and 2(method 1-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local space is madesmall, for instance, the length of each side is made half. For instance,the criteria is (18) or (18′) with respect to threshold values Tratioroior Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {Tratioroi}}{or}} & (18) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {Tdiffroi}},} & \left( 18^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (18) or (18′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 4: Condition for Terminating the Iterative Estimation of the 3DDisplacement Vector Distribution)

Below-described is the criteria for terminating the iterative estimationof the 3D displacement vector distribution. The processes 1, 2 and 3 ofmethod 1-2 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (19) or (19′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (19) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {aboveTdiffroi}},} & \left( 19^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

Final estimate is obtained from eq. (15) or eq. (16).

The initial estimate [eq. (1)] of the iterative estimation of the 3Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 1-3]

The flowchart of the method 1-3 is shown in FIG. 12. To prevent phasematching from diverging at the process 1 of the method 1-1, estimationerror is reduced of the residual vector. Possibility of divergence isdetected from above-described condition (14) or (14′), and byeffectively utilizing method 1-1 and 1-2, even if the SNR of theultrasound echo signals are low, accurate 3D displacement vectormeasurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 1-2 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

Phase matching and estimation of the 3D residual displacement vector areperformed at every point (x,y,z) in the 3D SOI. That is, the processes 1and 2 of the method 1-1 are performed once at every point in the SOI.Thus, the estimate of the 3D residual vector distribution is obtained[eq. (6-2)].

During this estimation, if neither condition (14) nor (14′) is satisfiedwith, the method 1-1 is used. If condition (14) or (14′) is satisfiedwith at points or spaces, in the process 2 of the method 1-2, oversufficiently large spaces centered on the points or spaces, or over theSOI, the estimate d^(i)(x,y,z) of the 3D displacement vector d(x,y,z)can be 3D low pass filtered or 3D median filtered as eq. (20).

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED[d ^(i)(x,y,z)]  (20)

Thus, the estimation error is reduced of the residual vector comparedwith process 2 of the method 1-1 [eq. (7)].

Thus, iterative estimation is terminated at the process 5 of the method1-1 or the process 4 of the method 1-2. Hence, final estimate isobtained from eq. (11), or eq. (15), or eq. (20).

The initial estimate [eq. (1)] of the iterative estimation of the 3Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 1-4]

The flowchart of the method 1-4 is shown in FIG. 13. To prevent phasematching from diverging at the process 1 of the method 1-1, estimationerror is reduced of the residual vector. Thus, even if the SNR of theultrasound echo signals are low, accurate 3D displacement vectormeasurement can be realized.

The procedure of iterative estimation is different from that of themethod 1-1. At i th estimate (i≧1), the following processes areperformed.

(Process 1: Estimation of the 3D Residual Displacement VectorDistribution)

Phase matching and estimation of the 3D residual displacement vector areperformed at every point (x,y,z) in the 3D SOI. That is, the process 1of the method 1-1 is performed once at every point in the SOI.

To obtain the estimate

^(i)(x,y,z)[=(

_(x) ^(i)(x,y,z),

_(y) ^(i)(x,y,z),

_(z) ^(i)(x,y,z))^(T)] of the residual vector distributionu^(i)(x,y,z)[=(u^(i) _(x)(x,y,z), u^(i) _(y)(x,y,z), u^(i)_(z)(x,y,z))^(T)], at every point local 3D echo cross-spectrum isevaluated from the 3D Fourier's transforms of the pre-deformation local3D ultrasound echo signal r₁(l,m,n) and phase-matched post-deformationlocal 3D ultrasound echo signal r^(i) ₂(l,m,n). Alternatively, whenpre-deformation local 3D ultrasound echo signal is phase-matched, atevery point cross-spectrum of r^(i) ₁(l,m,n) and r₂(l,m,n) is evaluated.

The least squares method is implemented on the gradient of the phasewith utilization of each weight function, i.e., the squaredcross-spectrum |S_(2,1) ^(i)(l,m,n)|², where each weight function isnormalized by the power of the cross-spectrum, i.e.,

$\sum\limits_{l,m,n}{{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}.}$Moreover, regularization method is also implemented. That is, byminimizing the next functional with respect to the vector u^(i)comprised of the 3D residual vector distribution u^(i)(x,y,z).

$\begin{matrix}{{{error}\left( u^{i} \right)} = {{{a - {Fu}^{i}}}^{2} + {\alpha_{li}{u^{i}}^{2}} + {\alpha_{2i}{{Gu}^{i}}^{2}} + {\alpha_{3i}{{G^{T}{Gu}^{i}}}^{2}} + {\alpha_{4i}{{{GG}^{T}{Gu}^{i}}}^{2}}}} & (21)\end{matrix}$

-   where a: vector comprised of (x,y,z) distribution of the    cross-spectrum phase Θ^(i)(l,m,n) weighted with cross-spectrum    |S_(2,1) ^(i)(l,m,n)| normalized by the magnitude of the    cross-spectrum

$\sqrt{\sum\limits_{l,m,n}}{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}$

-    evaluated at every point in the 3D SOI.    -   F: matrix comprised of (x,y,z) distribution of the Fourier's        coordinate value (l,m,n) weighted with cross-spectrum |S_(2,1)        ^(i)(l,m,n)| normalized by the magnitude of the cross-spectrum

$\sqrt{\sum\limits_{l,m,n}}{{S_{2,1}^{i}\left( {l,m,n} \right)}}^{2}$

-   -    evaluated at every point in the 3D SOI.    -   α_(1i), α_(2i), α_(3i), α_(4i): regularization parameter (at        least larger than zero)    -   Gu^(i): vector comprised of the finite difference approximations        of the 3D distributions of the 3D gradient components of the        unknown 3D residual vector u^(i)(x,y,z) components

${\frac{\partial}{\partial x}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial y}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial z}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial x}{u_{y}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial y}{u_{y}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial z}u_{y}^{i}\left( {x,y,z} \right)},{\frac{\partial}{\partial x}{u_{z}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial y}{u_{z}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial z}u_{z}^{i}\left( {x,y,z} \right)}$

-   -   G^(T)Gu^(i): vector comprised of the finite difference        approximations of the 3D distributions of the 3D Laplacians of        the unknown 3D residual vector u^(i)(x,y,z) components

${\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}$${\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}$${\frac{\partial^{2}}{\partial x^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{z}^{i}\left( {x,y,z} \right)}}$

-   -   GG^(T)Gu^(i): vector comprised of the finite difference        approximations of the 3D distributions of the 3D gradient        components of the 3D Laplacians of the unknown 3D residual        vector u^(i)(x,y,z) components

${\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial z}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial z}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{z}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{z}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial z}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{z}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{z}^{i}\left( {x,y,z} \right)}}} \right)},$

As ∥u^(i)∥², ∥Gu^(i)∥², ∥G^(T)Gu^(i)∥², ∥GG^(T)Gu^(i)∥² are positivedefinite, error(u^(i)) has one minimum value. Thus, by solving forresidual displacement vector distribution u^(i)(x,y,z) the simultaneousequations:(F ^(T) F+α _(1i) I+α _(2i) G ^(T) G+α _(3i) G ^(T) GG ^(T) G+α _(4i) G^(T) GG ^(T) GG ^(T) G)u ^(i) =F ^(T) a,  (22)estimate

^(i)(x,y,z)[=(

_(x) ^(i)(x,y,z),

_(y) ^(i)(x,y,z),

_(z) ^(i)(x,y,z))^(T)] of the residual vector distributionu^(i)(x,y,z)[=(u^(i) _(x)(x,y,z), u^(i) _(y)(x,y,z), u^(i)_(z)(x,y,z))^(T)] is stably obtained. Thus, estimation error is reducedof the residual vector.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameters depend on the correlation ofthe local echo data (peak value of the cross-correlation function,sharpness of the cross-correlation function, width of thecross-correlation function), the SNR of the cross-spectrum power, etc.;then position of the unknown displacement vector, direction of theunknown displacement component, direction of the partial derivative,etc.

(Process 2: Update of the Estimate of the 3D Displacement VectorDistribution)

The i−1 th estimate of the 3D displacement vector distribution isupdated using i th estimate of the 3D residual vector distribution.d ^(i)(x,y,z)=

^(i−1)(x,y,z)+

^(i)(x,y,z)  (23)

Freely, this estimate can be 3D low pass filtered or 3D median filter toyield the estimate of the 3D displacement vector distribution.

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED[d ^(i)(x,y,z)]  (24)

Hence, phase matching of the process 1 of method 1-4 is performed usingthe 3D residual vector data u^(i)(x,y,z) obtained from eq. (22), or the3D vector data d(x,y,z) obtained from eq. (23), or smoothed estimateobtained from eq. (24).

[Process 3: Condition for Heightening the Spatial Resolution of the 3DDisplacement Vector Distribution Measurement (Condition for Making theLocal Space Small)]

In order to make the spatial resolution high of the 3D displacementvector distribution measurement, during iterative estimation, the localspace used for each point is made small, or the local space used overthe SOI is made small.

The criteria for each point is below-described. The processes 1 and 2 ofmethod 1-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local space is madesmall, for instance, the length of each side is made half. For instance,the criteria is (25) or (25′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (25) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 25^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (25) or (25′) can be applied to each direction component,and in this case the side is shorten in each direction.

The criteria over the SOI is below-described. The processes 1 and 2 ofmethod 1-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local space is madesmall, for instance, the length of each side is made half. For instance,the criteria is (26) or (26′) with respect to threshold values Tratioroior Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {Tratioroi}}{or}} & (26) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {Tdiffroi}},} & \left( 26^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (26) or (26′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 4: Condition for Terminating the Iterative Estimation of the 3DDisplacement Vector Distribution)

Below-described is the criteria for terminating the iterative estimationof the 3D displacement vector distribution. The processes 1, 2 and 3 ofmethod 1-4 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (27) or (27′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (27) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {aboveTdiffroi}},} & \left( 27^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

Final estimate is obtained from eq. (23) or eq. (24).

The initial estimate [eq. (1)] of the iterative estimation of the 3Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 1-5]

The flowchart of the method 1-5 is shown in FIG. 14. To prevent phasematching from diverging at the process 1 of the method 1-1, estimationerror is reduced of the residual vector. Possibility of divergence isdetected from above-described condition (14) or (14′), and byeffectively utilizing method 1-1 and 1-4, even if the SNR of theultrasound echo signals are low, accurate 3D displacement vectormeasurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 1-4 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

Phase matching and estimation of the 3D residual displacement vector areperformed at every point (x,y,z) in the 3D SOI. That is, the process 1of the method 1-1 is performed once at every point in the SOI. Moreover,using the regularization method, stably the estimate of the 3D residualvector distribution is obtained.

During this estimation, if neither condition (14) nor (14′) is satisfiedwith, the method 1-1 is used. If condition (14) or (14′) is satisfiedwith at points or spaces, in the process 2 of the method 1-4, oversufficiently large spaces centered on the points or spaces, or over theSOI, the estimate d(x,y,z) of the 3D displacement vector d(x,y,z) can be3D low pass filtered or 3D median filtered as eq. (28).

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED [d ^(i)(x,y,z)]  (28)Thus, the estimation error is reduced of the residual vector.

Iterative estimation is terminated at the process 5 of the method 1-1 orthe process 4 of the method 1-4. Hence, final estimate is obtained fromeq. (11), or eq. (23), or eq. (28).

The initial estimate [eq. (1)] of the iterative estimation of the 3Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

(II) Method 2: Measurement of 2D Displacement Vector ComponentDistribution in 2D ROI

2D displacement vector distribution can be measured in 2D ROI 7 in theCartesian coordinate system. 2D ultrasound echo signals r₁(x,y) andr₂(x,y) are respectively acquired under pre-deformation andpost-deformation. These echo signals are processed by thebelow-described methods 2-1, 2-2, 2-3, 2-4, and 2-5. That is, as shownin FIG. 15, local region is set at each point in the pre- andpost-deformation 2D echo signal, and as shown in FIG. 16, thecorresponding local region is iteratively searched for in the ROI 7using the local phase characteristics as the index. In this searchingscheme, the estimated residual displacement vector is used to update thepreviously estimated displacement vector. When the estimated residualdisplacement vector satisfies with prescribed condition, the localregion size is made small (FIG. 17). Thus, accurate 2D displacementvector measurement is realized. Here, sampling intervals are Δx and Δyrespectively in the x and y-axes.

[Method 2-1]

The procedure of the method 2-1 is shown in FIG. 10. The processes from1 to 5 yields 2D displacement vector d(x,y)[=(dx(x,y), dy(x,y))^(T)] ofarbitrary point (x,y) in 2D ROI from pre- and post-deformation local 2Decho signals r₁(l,m) and r₂(l,m) [0≦l≦L−1, 0≦m≦M−1] centered on (x,y) ofpre- and post-deformation 2D echo signals r₁(x,y) and r₂(x,y). L and Mshould be determined such that ΔxL and ΔyM are respectively at least 4times longer than corresponding displacement components |dx(x,y)| and|dy(x,y)|.

(Process 1: Phase Matching at the Point (x,y))

Phase matching is performed to obtain i-th estimated^(i)(x,y)[=(d^(i)x(x,y), d^(i)y(x,y))^(T)] of the 2D displacementvector d(x,y)[=(dx(x,y), dy(x,y))^(T)].

Searching region is set in the post-deformation echo signal spacer₂(x,y), being centered on the local region [0≦l≦L−1, 0≦m≦M−1] centeredon (x,y) and being twice longer than the corresponding length, in orderto update the i−1 th estimate d^(i−1)(x,y)[=(d^(i−1)x(x,y), d^(i−1)y(x,y))^(T)] of the 2D displacement vector d(x,y)[=(dx(x,y), dy(x,y)^(T)),whered ⁰(x,y)={hacek over (d)}(x,y).  (29)

The phase of the post-deformation local echo signal is matched topre-deformation local echo signal by multiplying

$\begin{matrix}{\exp\left\{ {{j\frac{2\pi}{L}\frac{d_{x}^{i - 1}\left( {x,y} \right)}{\Delta\; x}l} + {j\frac{2\pi}{M}\frac{d_{y}^{i - 1}\left( {x,y} \right)}{\Delta\; y}m}} \right\}} & (30)\end{matrix}$to 2D Fourier's transform of this searching region echo signal r′₂(l,m)[0≦l≦2L−1, 0≦m≦2M−1] using i-th estimate d^(i−1)(x,y), or by multiplying

$\begin{matrix}{\exp\left\{ {{j\frac{2\pi}{L}\frac{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}{\Delta\; x}l} + {j\frac{2\pi}{M}\frac{{\overset{\Cap}{u}}_{y}^{i - 1}\left( {x,y} \right)}{\Delta\; y}m}} \right\}} & \left( 30^{\prime} \right)\end{matrix}$to 2D Fourier's transform of the i−1 th phase-matched searching regionecho signal r′^(i−1) ₂(l,m) using the estimate

^(i−1)(x,y)[=(

_(x) ^(i−1)(x,y),

_(y) ^(i−1)(x,y))^(T)] [

⁰(x,y)=0 (zero vector)] of the vector u^(i−1)(x,y)[=(u^(i−1) _(x)(x,y),u^(i−1) _(y)(x,y))^(T)].

By carrying out inverse Fourier's transform of this product,post-deformation echo signal r^(i) ₂(l,m) is obtained at the center ofthe searching region echo signal r′^(i) ₂(l,m), which is used at i-thstage to estimate 2D displacement vector d(x,y)[=(dx(x,y),dy(x,y))^(T)].

Alternatively, the phase of the pre-deformation local echo signal can bematched to post-deformation local echo signal in a similar way. That is,2D Fourier's transform of the searching region echo signal r′₁(l,m)[0≦l≦2L−1, 0≦m≦2M−1] centered on the point (x,y) in the pre-deformationecho signal region is multiplied with

$\begin{matrix}{{\exp\left\{ {{{- j}\frac{2\pi}{L}\frac{d_{x}^{i - 1}\left( {x,y} \right)}{\Delta\; x}l} - {j\frac{2\pi}{M}\frac{d_{y}^{i - 1}\left( {x,y} \right)}{\Delta\; y}m}} \right\}},} & \left( 30^{\prime\prime} \right)\end{matrix}$or 2D Fourier's transform of the i−1 th phase-matched searching regionecho signal r′^(i−1) ₁(l,m) is multiplied with

$\begin{matrix}{\exp{\left\{ {{{- j}\frac{2\pi}{L}\frac{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}{\Delta\; x}l} - {j\frac{2\pi}{M}\frac{{\overset{\Cap}{u}}_{y}^{i - 1}\left( {x,y} \right)}{\Delta\; y}m}} \right\}.}} & \left( 30^{\prime\prime\prime} \right)\end{matrix}$(Process 2: Estimation of 2D Residual Displacement Vector at the Point(x,y))

Local 2D echo cross-spectrum is evaluated from the 2D Fourier'stransforms of the pre-deformation local 2D ultrasound echo signalr₁(l,m) and phase-matched post-deformation local 2D ultrasound echosignal r^(i) ₂(l,m)S ^(i) _(2,1)(l,m)=R ₂ ^(i)*(l,m)R ₁(l,m),  (31)where * denotes conjugate.

Alternatively, when pre-deformation local 2D ultrasound echo signal isphase-matched, cross-spectrum of r^(i) ₁(l,m) and r₂(l,m) is evaluatedasS ^(i) _(2,1)(l,m)=R ₂*(l,m)R ^(i) ₁(l,m).Cross-spectrum is represented as

$\begin{matrix}{{{S_{2,1}^{i}\left( {l,m} \right)} \cong {{{R_{1}^{i}\left( {l,m} \right)}}^{2}\exp\left\{ {{j\frac{2\pi}{L}\frac{u_{x}^{i}\left( {x,y} \right)}{\Delta\; x}l} + {j\frac{2\pi}{M}\frac{u_{y}^{i}\left( {x,y} \right)}{\Delta\; y}}} \right\}}},} & (32)\end{matrix}$where 0≦l≦L−1, 0≦m≦M−1,and then the phase is represented as

$\begin{matrix}{{{\theta^{i}\left( {l,m} \right)} = {\tan^{- 1}\left( \frac{{Im}\left\lbrack {S_{2,1}^{i}\left( {l,m} \right)} \right\rbrack}{{Re}\left\lbrack {S_{2,1}^{i}\left( {l,m} \right)} \right\rbrack} \right)}},} & (33)\end{matrix}$where Re[•] and Im[•] respectively represent the real and imaginarycomponent of “•”.

The least squares method is implemented on the gradient of the phase eq.(33) weighted with squared cross-spectrum|S _(2,1) ^(i)(l,m)|²(=Re ² [S _(2,1) ^(i)(l,m)]² +Im ² [S _(2,1)^(i)(l,m]).That is, by minimizing functional:

error (u^(i)(x,y))

$\begin{matrix}{= {\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( {{\theta^{i}\left( {l,m} \right)} - {{u_{x}^{i}\left( {x,y} \right)}\left( \frac{2\pi}{L\;\Delta\; x} \right)l} - {{u_{y}^{i}\left( {x,y} \right)}\left( \frac{2\pi}{M\;\Delta\; y} \right)m}} \right)^{2}}}} & (34)\end{matrix}$with respect to the 2D residual vector u^(i)(x,y) to be used to updatethe i−1 th estimate d^(i−1)(x,y) of the 2D displacement vector d(x,y),the estimate of u^(i)(x,y) is obtained as

^(i)(x,y)[=(

_(x) ^(i)(x,y),

_(y) ^(i)(x,y))^(T)].Concretely, the next simultaneous equations are solved.

$\begin{matrix}{\begin{bmatrix}{\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)l\;{\theta^{i}\left( {l,m} \right)}}} \\{\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( \frac{2\pi}{M\;\Delta\; y} \right)m\;{\theta^{i}\left( {l,m} \right)}}}\end{bmatrix} = {\begin{bmatrix}{\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)^{2}l^{2}}} & {\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)\left( \frac{2\pi}{M\;\Delta\; y} \right){lm}}} \\{\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)\left( \frac{2\;\pi}{M\;\Delta\; y} \right){lm}}} & {\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}\left( \frac{2\pi}{M\;\Delta\; y} \right)^{2}m^{2}}}\end{bmatrix} \times {\quad\begin{bmatrix}{u_{x}^{i}\left( {x,y} \right)} \\{u_{y}^{i}\left( {x,y} \right)}\end{bmatrix}}}} & (35)\end{matrix}$

When the 2D displacement vector d(x,y) is large, the 2D residualdisplacement vector u^(i)(x,y) needs to be estimated after unwrappingthe phase of the cross-spectrum [eq. (31)] in the frequency domain(l,m).

Alternatively, when the 2D displacement vector d(x,y) is large, by usingcross-correlation method (evaluation of the peak position of thecross-correlation function obtained as 2D inverse Fourier's transform ofthe cross-spectrum [eq. (31)]) at the initial stages during iterativeestimation, the 2D residual displacement vector u^(i)(x,y) can beestimated without unwrapping the phase of the cross-spectrum [eq. (31)]in the frequency domain. Specifically, by using the cross-correlationmethod, x and y components of the 2D displacement vector arerespectively estimated as integer multiplications of the ultrasound echosampling intervals Δx, Δy. For instance, with respect to thresholdvalues correTratio or correTdiff, after

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \leq {correTratio}}{or}} & (36) \\{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}} \leq {correTdiff}} & \left( 36^{\prime} \right)\end{matrix}$is satisfied with where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i-1 thestimates of the residual vectors, by using the estimate of the 2Ddisplacement vector d(x,y) as the initial estimate, the 2D residualdisplacement vector is estimated from the gradient of the phase of thecross-spectrum [eq. (31)].

Empirically it is known that after using cross-correlation method theconditions |u^(i) _(x)(x,y)|≦Δx/2, |u^(i) _(y)(x,y)|≦Δy/2 are satisfiedwith. Then, the necessary and sufficient condition for allowingestimation of the 2D residual displacement vector without unwrapping thephase of the cross-spectrum

$\begin{matrix}{{{\frac{u_{x}^{i}\left( {x,y} \right)}{\Delta\; x} + \frac{u_{y}^{i}\left( {x,y} \right)}{\Delta\; y}}} \leq 1} & (37)\end{matrix}$is satisfied with.

Alternatively, when the magnitude of the 2D displacement vector d(x,y)is large, at initial stages, the acquired original ultrasound echo datacan be thinned out with constant interval in each direction and thereduced echo data can be used such that the 2D residual displacementvector can be estimated without unwrapping the phase of thecross-spectrum [eq. (31)] in the frequency domain (l,m). Specifically,the acquired original ultrasound echo data are thinned out with constantinterval in each direction and the reduced echo data are used such thatthe condition (37) or (37′) is satisfied with.|u _(x) ^(i)(x,y)|≦Δx/2 and |u _(y) ^(i)(x,y)|≦Δy/2.  (37′)

The iteration number i increasing, i.e., the magnitude of the 2Dresidual displacement vector components u^(i) _(x)(x,y), u^(i) _(y)(x,y)decreasing, the ultrasound echo data densities are made restored in eachdirection, for instance, twice per iteration.

The interval of the ultrasound echo signal data are shortened, forinstance, when with respect to threshold values stepTratio or stepTdiffthe condition

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \leq {stepTratio}}{or}} & (38) \\{{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}} \leq {stepTdiff}} & \left( 38^{\prime} \right)\end{matrix}$is satisfied with, where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

The condition (38) or (38′) can be applied to each direction component,and in this case the data interval is shorten in each direction. Theseare also applied to below-described methods 2-2, 2-3, 2-4, and 2-5.

(Process 3: Update of the 2D Displacement Vector Estimate of the Point(x,y))

Thus, the i th estimate of the 2D displacement vector d(x,y) isevaluated asd ^(i)(x,y)=d ^(i−1)(x,y)+

^(i)(x,y)  (39)[Process 4: Condition for Heightening the Spatial Resolution of the 2DDisplacement Vector Distribution Measurement (Condition for Making theLocal Region Small)]

In order to make the spatial resolution high of the 2D displacementvector distribution measurement, the local region is made small duringiterative estimation. The criteria is below-described. The processes 1,2 and 3 are iteratively carried out till the criteria is satisfied with.When the criteria is satisfied with, the local region is made small, forinstance, the length of each side is made half. For instance, thecriteria is (40) or (40′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \leq {Tratio}}{or}} & (40) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}} \leq {Tdiff}},} & \left( 40^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.The condition (40) or (40′) can be applied to each direction component,and in this case the side is shorten in each direction.(Process 5: Condition for Terminating the Iterative Estimation of the 2DDisplacement Vector of the Point (x,y))

Below-described is the criteria for terminating the iterative estimationof the 2D displacement vector of each point. The processes 1, 2 and 3are iteratively carried out till the criteria is satisfied with. Forinstance, the criteria is (41) or (41′) with respect to threshold valuesaboveTratio or aboveTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \leq {aboveTratio}}{or}} & (41) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}} \leq {aboveTdiff}},} & \left( 41^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.(Process 6)

The 2D displacement vector component distributions are obtained bycarrying out processes 1, 2, 3, 4, and 5 at every point in the 2D ROI.

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Limitation of Method 2-1]

The estimate of the 2D displacement vector d(x,y) is iteratively updatedat each point (x,y) in the 2D ROI. Being dependent on the SNR of thelocal 2D echo signals, particularly at initial stages errors possiblyoccur when estimating the residual vector and then phase matchingpossibly diverges. For instance, when solving eq. (35) [process 2] ordetecting the peak position of the cross-correlation function [process2], errors possibly occur.

The possibility for divergence of the phase matching is, for instance,confirmed by the condition (42) or (42′) with respect to the thresholdvalue belowTratio or BelowTdiff.

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \geq {belowTratio}} & (42) \\{or} & \; \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}} \geq {belowTdiff}},} & \left( 42^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

To prevent phase matching (process 1) from diverging, in thebelow-described methods 2-2, 2-3, 2-4, and 2-5, by freely using thecondition (42) or (42′), estimation error is reduced of the residualvector. Thus, even if the SNR of the ultrasound echo signals are low,accurate 2D displacement vector measurement can be realized.

[Method 2-2]

The flowchart of the method 2-2 is shown in FIG. 11. To prevent phasematching from diverging at the process 1 of the method 2-1, estimationerror is reduced of the residual vector. Thus, even if the SNR of theultrasound echo signals are low, accurate 2D displacement vectormeasurement can be realized.

The procedure of iterative estimation is different from that of themethod 2-1. At i th estimate (i≧1), the following processes areperformed.

(Process 1: Estimation of the 2D Residual Displacement VectorDistribution)

Phase matching and estimation of the 2D residual displacement vector areperformed at every point (x,y) in the 2D ROI. That is, the processes 1and 2 of the method 2-1 are performed once at every point in the ROI.Thus, the estimate of the 2D residual vector distribution is obtained.

(Process 2: Update of the Estimate of the 2D Displacement VectorDistribution)

The i−1 th estimate of the 2D displacement vector distribution isupdated using i th estimate of the 2D residual vector distribution.d ^(i)(x,y)=

^(i−1)(x,y)+

^(i)(x,y)  (43)

Next, this estimate is 2D low pass filtered or 2D median filter to yieldthe estimate of the 2D displacement vector distribution:

^(i)(x,y)=LPF[d ^(i)(x,y)], or

^(i)(x,y)=MED[d ^(i)(x,y)]  (44)

Thus, the estimation error is reduced of the residual vector comparedwith process 2 of the method 2-1 [eq. (35)]. Hence, phase matching ofthe process 1 of method 2-2 is performed using smoothed estimate of the2D displacement vector distribution.

[Process 3: Condition for Heightening the Spatial Resolution of the 2DDisplacement Vector Distribution Measurement (Condition for Making theLocal Region Small)]

In order to make the spatial resolution high of the 2D displacementvector distribution measurement, during iterative estimation, the localregion used for each point is made small, or the local region used overthe ROI is made small.

The criteria for each point is below-described. The processes 1 and 2(method 2-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (45) or (45′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \leq {Tratio}} & (45) \\{or} & \; \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}} \leq {Tdiff}},} & \left( 45^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

The condition (45) or (45′) can be applied to each direction component,and in this case the side is shorten in each direction.

The criteria over the ROI is below-described. The processes 1 and 2(method 2-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (46) or (46′) with respect to threshold values Tratioroior Tdiffroi.

$\begin{matrix}{\frac{\sum\limits_{{({x,y})}\mspace{11mu} \in {ROI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})}\mspace{11mu} \in {ROI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {Tratioroi}} & (46) \\{or} & \; \\{{{\sum\limits_{{({x,y})}\mspace{11mu} \in {ROI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}}} \leq {Tdiffroi}},} & \left( 46^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

The condition (46) or (46′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 4: Condition for Terminating the Iterative Estimation of the 2DDisplacement Vector Distribution)

Below-described is the criteria for terminating the iterative estimationof the 2D displacement vector distribution. The processes 1, 2 and 3 ofmethod 2-2 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (47) or (47′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{\frac{\sum\limits_{{({x,y})}\mspace{11mu} \in {ROI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})}\mspace{11mu} \in {ROI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {aboveTratioroi}} & (47) \\{or} & \; \\{{{\sum\limits_{{({x,y})}\mspace{11mu} \in {ROI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}}} \leq {aboveTdiffroi}},} & \left( 47^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

Final estimate is obtained from eq. (43) or eq. (44).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 2-3]

The flowchart of the method 2-3 is shown in FIG. 12. To prevent phasematching from diverging at the process 1 of the method 2-1, estimationerror is reduced of the residual vector. Possibility of divergence isdetected from above-described condition (42) or (42′), and byeffectively utilizing method 2-1 and 2-2, even if the SNR of theultrasound echo signals are low, accurate 2D displacement vectormeasurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 2-2 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

Phase matching and estimation of the 2D residual displacement vector areperformed at every point (x,y) in the 2D ROI. That is, the processes 1and 2 of the method 2-1 are performed once at every point in the ROI.Thus, the estimate of the 2D residual vector distribution is obtained.

During this estimation, if neither condition (42) nor (42′) is satisfiedwith, the method 2-1 is used. If condition (42) or (42′) is satisfiedwith at points or regions, in the process 2 of the method 2-2, oversufficiently large regions centered on the points or regions, or overthe ROI, the estimate d^(i)(x,y) of the 2D displacement vector d(x,y)can be 2D low pass filtered or 2D median filtered as eq. (48).

^(i)(x,y)=LPF[d ^(i)(x,y)], or

^(i)(x,y)=MED[d ^(i)(x,y)]  (48)

Thus, the estimation error is reduced of the residual vector comparedwith process 2 of the method 2-1 [eq. (35)].

Thus, iterative estimation is terminated at the process 5 of the method2-1 or the process 4 of the method 2-2. Hence, final estimate isobtained from eq. (39), or eq. (43), or eq. (48).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 2-4]

The flowchart of the method 2-4 is shown in FIG. 13. To prevent phasematching from diverging at the process 1 of the method 2-1, estimationerror is reduced of the residual vector. Thus, even if the SNR of theultrasound echo signals are low, accurate 2D displacement vectormeasurement can be realized.

The procedure of iterative estimation is different from that of themethod 2-1. At i th estimate (i≧1), the following processes areperformed.

(Process 1: Estimation of the 2D Residual Displacement VectorDistribution)

Phase matching and estimation of the 2D residual displacement vector areperformed at every point (x,y) in the 2D ROI. That is, the process 1 ofthe method 2-1 is performed once at every point in the ROI.

To obtain the estimate

^(i)(x,y)[=(

_(x) ^(i)(x,y),

_(y) ^(i)(x,y))^(T)] of the residual vector distributionu^(i)(x,y)[=(u^(i) _(x)(x,y),u^(i) _(y)(x,y))^(T)], at every point local2D echo cross-spectrum is evaluated from the 2D Fourier's transforms ofthe pre-deformation local 2D ultrasound echo signal r₁(l,m) andphase-matched post-deformation local 2D ultrasound echo signal r^(i)₂(l,m) Alternatively, when pre-deformation local 2D ultrasound echosignal is phase-matched, at every point cross-spectrum of r^(i) ₁(l,m)and r₂(l,m) is evaluated.

The least squares method is implemented on the gradient of the phasewith utilization of each weight function, i.e., the squaredcross-spectrum |S_(2,1) ^(i)(l,m)|², where each weight function isnormalized by the power of the cross-spectrum, i.e.,

$\sum\limits_{l,m}{{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}.}$Moreover, regularization method is also implemented. That is, byminimizing the next functional with respect to the vector u^(i)comprised of the 2D residual vector distribution u^(i)(x,y).

$\begin{matrix}\begin{matrix}{{{error}\mspace{11mu}\left( u^{i} \right)} = {{{a - {Fu}^{i}}}^{2} + {\alpha_{1i}{u^{i}}^{2}} + {\alpha_{2i}{{Gu}^{i}}^{2}} +}} \\{{\alpha_{3i}{{G^{T}{Gu}^{i}}}^{2}} + {\alpha_{4i}{{{GG}^{T}{Gu}^{i}}}^{2}}}\end{matrix} & (49)\end{matrix}$

-   where a: vector comprised of (x,y) distribution of the    cross-spectrum phase Θ^(i)(l,m) weighted with cross-spectrum    |S_(2,1) ^(i)(l,m)| normalized by the magnitude of the    cross-spectrum

$\sqrt{\;}{\sum\limits_{l,m}{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}}$

-    evaluated at every point in the 2D ROI.    -   F: matrix comprised of (x,y) distribution of the Fourier's        coordinate value (l,m) weighted with cross-spectrum |S_(2,1)        ^(i)(l,m)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l,m}{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}}$

-   -    evaluated at every point in the 2D ROI.    -   α_(1i), α_(2i), α_(3i), α_(4i): regularization parameter (at        least larger than zero)    -   Gu^(i): vector comprised of the finite difference approximations        of the 2D distributions of the 2D gradient components of the        unknown 2D residual vector u^(i)(x,y) components

${\frac{\partial}{\partial x}{u_{x}^{i}\left( {x,y} \right)}},{\frac{\partial}{\partial y}{u_{x}^{i}\left( {x,y} \right)}},{\frac{\partial}{\partial x}{u_{y}^{i}\left( {x,y} \right)}},{\frac{\partial}{\partial y}{u_{y}^{i}\left( {x,y} \right)}}$

-   -   G^(T)Gu^(i): vector comprised of the finite difference        approximations of the 2D distributions of the 2D Laplacians of        the unknown 2D residual vector u^(i)(x,y) components

${\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y} \right)}}$${\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y} \right)}}$

-   -   GG^(T)Gu^(i): vector comprised of the finite difference        approximations of the 2D distributions of the 2D gradient        components of the 2D Laplacians of the unknown 2D residual        vector u^(i)(x,y) components

${\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y} \right)}}} \right)},{\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y} \right)}}} \right)}$

As ∥u^(i)∥², ∥Gu^(i)∥², ∥G^(T)Gu^(i)∥², ∥GG^(T)Gu^(i)∥² are positivedefinite, error(u^(i)) has one minimum value. Thus, by solving forresidual displacement vector distribution u^(i)(x,y) the simultaneousequations:(F ^(T) F+α _(1i) I+α _(2i) G ^(T) G+α _(3i) G ^(T) GG ^(T) G+α _(4i) G^(T) GG ^(T) GG ^(T) G)u ^(i) =F ^(T) a,  (50)estimate

^(i)(x,y)[=(

_(x) ^(i)(x,y),

_(y) ^(i)(x,y))^(T)] of the residual vector distributionu^(i)(x,y)[=(u^(i) _(x)(x,y), u^(i) _(y)(x,y))^(T)] is stably obtained.Thus, estimation error is reduced of the residual vector.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameters depend on the correlation ofthe local echo data (peak value of the cross-correlation function,sharpness of the cross-correlation function, width of thecross-correlation function), the SNR of the cross-spectrum power, etc.;then position of the unknown displacement vector, direction of theunknown displacement component, direction of the partial derivative,etc.

(Process 2: Update of the Estimate of the 2D Displacement VectorDistribution)

The i−1 th estimate of the 2D displacement vector distribution isupdated using i th estimate of the 2D residual vector distribution.d ^(i)(x,y)=

^(i−1)(x,y)+

^(i)(x,y)  (51)

Freely, this estimate can be 2D low pass filtered or 2D median filter toyield the estimate of the 2D displacement vector distribution.

^(i)(x,y)=LPF[d ^(i)(x,y)], or

^(i)(x,y)=MED [d ^(i)(x,y)]  (52)Hence, phase matching of the process 1 of method 2-4 is performed usingthe 2D residual vector data u^(i)(x,y) obtained from eq. (50), or the 2Dvector data d^(i)(x,y) obtained from eq. (51), or smoothed estimateobtained from eq. (52). [Process 3: Condition for Heightening theSpatial Resolution of the 2D Displacement Vector DistributionMeasurement (Condition for Making the Local Region Small)]

In order to make the spatial resolution high of the 2D displacementvector distribution measurement, during iterative estimation, the localregion used for each point is made small, or the local region used overthe ROI is made small.

The criteria for each point is below-described. The processes 1 and 2 ofmethod 2-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (25) or (25′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}} \leq {Tratio}}{or}} & (53) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}} \leq {Tdiff}},} & \left( 53^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

The condition (53) or (53′) can be applied to each direction component,and in this case the side is shorten in each direction.

The criteria over the ROI is below-described. The processes 1 and 2 ofmethod 2-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (54) or (54′) with respect to threshold values Tratioroior Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {Tratioroi}}{or}} & (54) \\{{{\sum\limits_{{({x,y})} \in {ROI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}}} \leq {Tdiffroi}},} & \left( 54^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i-1 thestimates of the residual vectors.

The condition (54) or (54′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 4: Condition for Terminating the Iterative Estimation of the 2DDisplacement Vector Distribution)

Below-described is the criteria for terminating the iterative estimationof the 2D displacement vector distribution. The processes 1, 2 and 3 ofmethod 2-4 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (55) or (55′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (55) \\{{{\sum\limits_{{({x,y})} \in {ROI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y} \right)}}}} \leq {aboveTdiffroi}},} & \left( 55^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y)∥ and ∥

^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th and i−1 thestimates of the residual vectors.

Final estimate is obtained from eq. (51) or eq. (52).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 2-5]

The flowchart of the method 2-5 is shown in FIG. 14. To prevent phasematching from diverging at the process 1 of the method 2-1, estimationerror is reduced of the residual vector. Possibility of divergence isdetected from above-described condition (42) or (42′), and byeffectively utilizing method 2-1 and 2-4, even if the SNR of theultrasound echo signals are low, accurate 2D displacement vectormeasurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 2-4 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

Phase matching and estimation of the 2D residual displacement vector areperformed at every point (x,y) in the 2D ROI. That is, the process 1 ofthe method 2-1 is performed once at every point in the ROI. Moreover,using the regularization method, stably the estimate of the 2D residualvector distribution is obtained.

i−1 th estimate

^(i−1)(x,y) of 2D displacement vector distribution d(x,y).

i th estimate û^(i)(x,y) of 2D residual vector distribution u^(i)(x, y).

During this estimation, if neither condition (42) nor (42′) is satisfiedwith, the method 2-1 is used. If condition (42) or (42′) is satisfiedwith at points or regions, in the process 2 of the method 2-4, oversufficiently large regions centered on the points or regions, or overthe ROI, the estimate d(x,y) of the 2D displacement vector d(x,y) can be2D low pass filtered or 2D median filtered as eq. (56).

^(i)(x,y)=LPF[d ^(i)(x,y)], or

^(i)(x,y)=MED[d ^(i)(x,y)]  (56)Thus, the estimation error is reduced of the residual vector.

Iterative estimation is terminated at the process 5 of the method 2-1 orthe process 4 of the method 2-4. Hence, final estimate is obtained fromeq. (39), or eq. (51), or eq. (56).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

(III) Method 3: Measurement of 1D (One Direction) Displacement ComponentDistribution in 1D ROI

1D displacement component distribution can be measured in 1D ROI 7 inthe Cartesian coordinate system. 1D ultrasound echo signals r₁(x) andr₂(x) are respectively acquired under pre-deformation andpost-deformation. These echo signals are processed by thebelow-described methods 3-1, 3-2, 3-3, 3-4, and 3-5. That is, as shownin FIG. 18, local region is set at each point in the pre- andpost-deformation 1D echo signal, and as shown in FIG. 19, thecorresponding local region is iteratively searched for in the ROI 7using the local phase characteristics as the index. In this searchingscheme, the estimated residual displacement component is used to updatethe previously estimated displacement component. When the estimatedresidual displacement component satisfies with prescribed condition, thelocal region size is made small (FIG. 20). Thus, accurate 1Ddisplacement component measurement is realized. Here, sampling intervalis Δx in the x-axis.

[Method 3-1]

The procedure of the method 3-1 is shown in FIG. 10. The processes from1 to 5 yields 1D displacement component dx(x) of arbitrary point x in 1DROI from pre- and post-deformation local 1D echo signals r₁(l) and r₂(l)[0≦l≦L−1] centered on x of pre- and post-deformation 1D echo signalsr₁(x) and r₂(x). L should be determined such that ΔxL is at least 4times longer than the displacement component |dx(x)|.

(Process 1: Phase Matching at the Point x)

Phase matching is performed to obtain i-th estimate d^(i)x(x) of the 1Ddisplacement component dx(x).

Searching region is set in the post-deformation echo signal space r₂(x),being centered on the local region [0≦l≦L−1] centered on x and beingtwice longer than the local region length, in order to update the i−1 thestimate d^(i−1)x(x) of the 1D displacement component dx(x), wheredx ⁰(x)={hacek over (d)}x(x).  (57)

The phase of the post-deformation local echo signal is matched topre-deformation local echo signal by multiplying

$\begin{matrix}{\exp\left\{ {j\frac{2\pi}{L}\frac{d_{x}^{i - 1}(x)}{\Delta\; x}l} \right\}} & (58)\end{matrix}$to 1D Fourier's transform of this searching region echo signal r′₂(l)[0≦l≦2L−1] using i-th estimate dx^(i−1)(x), or by multiplying

$\begin{matrix}{\exp\left\{ {j\frac{2\pi}{L}\frac{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}{\Delta\; x}l} \right\}} & \left( 58^{\prime} \right)\end{matrix}$to 1D Fourier's transform of the i−1 th phase-matched searching regionecho signal r′^(i−1) ₂(l) using the estimate

_(x) ^(i−1)(x)[

_(x) ⁰(x)=0 (zero)] of the component u^(i−1) _(x)(x).

By carrying out inverse Fourier's transform of this product,post-deformation echo signal r^(i) ₂(l) is obtained at the center of thesearching region echo signal r′^(i) ₂(l), which is used at i-th stage toestimate 1D displacement component dx(x).

Alternatively, the phase of the pre-deformation local echo signal can bematched to post-deformation local echo signal in a similar way. That is,1D Fourier's transform of the searching region echo signal r′₁(l)[0≦l≦2L−1] centered on the point x in the pre-deformation echo signalregion is multiplied with

$\begin{matrix}{{\exp\left\{ {{- j}\frac{2\pi}{L}\frac{d_{x}^{i - 1}(x)}{\Delta\; x}l} \right\}},} & \left( 58^{\prime\prime} \right)\end{matrix}$or 1D Fourier's transform of the i−1 th phase-matched searching regionecho signal r′^(i−1) ₁(l) is multiplied with

$\begin{matrix}{\exp{\left\{ {{- j}\frac{2\pi}{L}\frac{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}{\Delta\; x}l} \right\}.}} & \left( 58^{\prime\prime\prime} \right)\end{matrix}$(Process 2: Estimation of 1D Residual Displacement Component at thePoint x)

Local 1D echo cross-spectrum is evaluated from the 1D Fourier'stransforms of the pre-deformation local 1D ultrasound echo signal r₁(l)and phase-matched post-deformation local 1D ultrasound echo signal r^(i)₂(l)S ^(i) _(2,1)(l)=R ₂ ^(i)*(l)R ₁(l),  (59)where * denotes conjugate.

Alternatively, when pre-deformation local 1D ultrasound echo signal isphase-matched, cross-spectrum of r^(i) ₁(l) and r₂(l) is evaluated asS ^(i) _(2,1)(l)=R ₂*(l)R ^(i) ₁(l)Cross-spectrum is represented as

$\begin{matrix}{{{S_{2,1}^{i}(l)} \cong {{{R_{1}^{i}(l)}}^{2}\exp\left\{ {j\frac{2\;\pi}{L}\frac{u_{x}^{i}(x)}{\Delta\; x}l} \right\}}},} & (60)\end{matrix}$where 0≦l≦L−1,and then the phase is represented as

$\begin{matrix}{{{\theta^{i}(l)} = {\tan^{- 1}\left( \frac{{Im}\left\lbrack {S_{2,1}^{i}(l)} \right\rbrack}{{Re}\left\lbrack {S_{2,1}^{i}(l)} \right\rbrack} \right)}},} & (61)\end{matrix}$where Re[•] and Im[•] respectively represent the real and imaginarycomponent of “•”.

The least squares method is implemented on the gradient of the phase eq.(61) weighted with squared cross-spectrum |S_(2,1)^(i)(l)|²(=Re²[S_(2,1) ^(i)(l)]²+Im²[S_(2,1) ^(i)(l)]). That is, byminimizing functional:

$\begin{matrix}{{{error}\mspace{14mu}\left( {u_{x}^{i}(x)} \right)} = {\sum\limits_{l}{{{S_{2,1}^{i}(l)}}^{2}\left( {{\theta^{i}(l)} - {{u_{x}^{i}(x)}\left( \frac{2\pi}{L\;\Delta\; x} \right)l}} \right)^{2}}}} & (62)\end{matrix}$with respect to the 1D residual component u_(x) ^(i)(x) to be used toupdate the i−1 th estimate dx^(i−1)(x) of the 1D displacement componentdx(x), the estimate of u_(x) ^(i)(x) is obtained as

_(x) ^(i)(x). Concretely, the next equation is solved.

$\begin{matrix}{{\sum\limits_{l}{{{S_{2,1}^{i}(l)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)l\;{\theta^{i}(l)}}} = {\sum\limits_{l}{{{S_{2,1}^{i}(l)}}^{2}\left( \frac{2\pi}{L\;\Delta\; x} \right)^{2}l^{2}{u_{x}^{i}(x)}}}} & (63)\end{matrix}$

When the 1D displacement component dx(x) is large, the 1D residualdisplacement component u_(x) ^(i)(x) needs to be estimated afterunwrapping the phase of the cross-spectrum [eq. (59)] in the frequencydomain 1.

Alternatively, when the 1D displacement component dx(x) is large, byusing cross-correlation method (evaluation of the peak position of thecross-correlation function obtained as 1D inverse Fourier's transform ofthe cross-spectrum [eq. (59)]) at the initial stages during iterativeestimation, the 1D residual displacement component u_(x) ^(i)(x) can beestimated without unwrapping the phase of the cross-spectrum [eq. (59)]in the frequency domain. Specifically, by using the cross-correlationmethod, x component of the 1D displacement component is estimated asinteger multiplication of the ultrasound echo sampling interval Δx. Forinstance, with respect to threshold values correTratio or correTdiff,after

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \leq {correTratio}} & (64) \\{or} & \; \\{{{{\overset{\Cap}{u}}_{x}^{i}(x)}} \leq {correTdiff}} & \left( 64^{\prime} \right)\end{matrix}$is satisfied with where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components, by using the estimate of the 1Ddisplacement component dx(x) as the initial estimate, the 1D residualdisplacement component is estimated from the gradient of the phase ofthe cross-spectrum [eq. (59)].

Empirically it is known that after using cross-correlation method thecondition |u^(i) _(x)(x)|≦Δx/2 is satisfied with. Then, the necessaryand sufficient condition for allowing estimation of the 1D residualdisplacement component without unwrapping the phase of thecross-spectrum

$\begin{matrix}{{\frac{u_{x}^{i}(x)}{\Delta\; x}} \leq 1} & (65)\end{matrix}$is satisfied with.

Alternatively, when the magnitude of the 1D displacement component dx(x)is large, at initial stages, the acquired original ultrasound echo datacan be thinned out with constant interval in the direction and thereduced echo data can be used such that the 1D residual displacementcomponent can be estimated without unwrapping the phase of thecross-spectrum [eq. (59)] in the frequency domain l. Specifically, theacquired original ultrasound echo data are thinned out with constantinterval in the direction and the reduced ehco data are used such thatthe condition (65) or (65′) is satisfied with.|u _(x) ^(i)(x)|≦Δx/2  (65′)

The iteration number i increasing, i.e., the magnitude of the 1Dresidual displacement component u^(i) _(x)(x) decreasing, the ultrasoundecho data density is made restored in the direction, for instance, twiceper iteration.

The interval of the ultrasound echo signal data are shortened, forinstance, when with respect to threshold values stepTratio or stepTdiffthe condition

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \leq {stepTratio}} & (66) \\{or} & \; \\{{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}} \leq {stepTdiff}} & \left( 66^{\prime} \right)\end{matrix}$is satisfied with, where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.(Process 3: Update of the 1D Displacement Component Estimate of thePoint x)

Thus, the i th estimate of the 1D displacement component dx(x) isevaluated asdx ^(i)(x)=dx ^(i−1)(x)+

_(x) ^(i)(x).  (67)[Process 4: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement (Condition for Makingthe Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement, the local region is made smallduring iterative estimation. The criteria is below-described. Theprocesses 1, 2 and 3 are iteratively carried out till the criteria issatisfied with. When the criteria is satisfied with, the local region ismade small, for instance, the length of the local region is made half.For instance, the criteria is (68) or (68′) with respect to thresholdvalues Tratio or Tdiff.

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \leq {Tratio}} & (68) \\{\;{or}} & \; \\{{{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}} \leq {Tdiff}},} & \left( 68^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.(Process 5: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component of the Point x)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component of each point. The processes 1, 2 and 3are iteratively carried out till the criteria is satisfied with. Forinstance, the criteria is (69) or (69′) with respect to threshold valuesaboveTratio or aboveTdiff.

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \leq {aboveTratio}} & (69) \\{or} & \; \\{{{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}} \leq {aboveTdiff}},} & \left( 69^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.(Process 6)

The 1D displacement component distribution is obtained by carrying outprocesses 1, 2, 3, 4, and 5 at every point in the 1D ROI.

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Limitation of Method 3-1]

The estimate of the 1D displacement component dx(x) is iterativelyupdated at each point x in the 1D ROI. Being dependent on the SNR of thelocal 1D echo signals, particularly at initial stages errors possiblyoccur when estimating the residual component and then phase matchingpossibly diverges. For instance, when solving eq. (63) [process 2] ordetecting the peak position of the cross-correlation function [process2], errors possibly occur.

The possibility for divergence of the phase matching is, for instance,confirmed by the condition (70) or (70′) with respect to the thresholdvalue belowTratio or BelowTdiff.

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \geq {belowTratio}} & (70) \\{or} & \; \\{{{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}} \geq {belowTdiff}},} & \left( 70^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.

To prevent phase matching (process 1) from diverging, in thebelow-described methods 3-2, 3-3, 3-4, and 3-5, by freely using thecondition (70) or (70′), estimation error is reduced of the residualcomponent. Thus, even if the SNR of the ultrasound echo signals are low,accurate 1D displacement component measurement can be realized.

[Method 3-2]

The flowchart of the method 3-2 is shown in FIG. 11. To prevent phasematching from diverging at the process 1 of the method 3-1, estimationerror is reduced of the residual component. Thus, even if the SNR of theultrasound echo signals are low, accurate 1D displacement componentmeasurement can be realized.

The procedure of iterative estimation is different from that of themethod 3-1. At i th estimate (i≧1), the following processes areperformed.

(Process 1: Estimation of the 1D Residual Displacement ComponentDistribution)

Phase matching and estimation of the 1D residual displacement componentare performed at every point x in the 1D ROI. That is, the processes 1and 2 of the method 3-1 are performed once at every point in the ROI.Thus, the estimate of the 1D residual component distribution isobtained.

(Process 2: Update of the Estimate of the 1D Displacement ComponentDistribution)

The i−1 th estimate of the 1D displacement component distribution isupdated using i th estimate of the 1D residual component distribution.dx ^(i)(x)=

_(x) ^(i−1)(x)+

_(x) ^(i)(x)  (71)

Next, this estimate is 1D low pass filtered or 1D median filter to yieldthe estimate of the 1D displacement component distribution:

_(x) ^(i)(x)=LPF[dx ^(i)(x)], or

_(x) ^(i)(x)=MED[dx ^(i)(x)].  (72)

Thus, the estimation error is reduced of the residual component comparedwith process 2 of the method 3-1 [eq. (63)]. Hence, phase matching ofthe process 1 of method 3-2 is performed using smoothed estimate of the1D displacement component distribution.

[Process 3: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement (Condition for Makingthe Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement, during iterative estimation, thelocal region used for each point is made small, or the local region usedover the ROI is made small.

The criteria for each point is below-described. The processes 1 and 2(method 3-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (73) or (73′) with respect to threshold valuesTratio or Tdiff.

$\begin{matrix}{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \leq {Tratio}} & (73) \\{or} & \; \\{{{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}} \leq {Tdiff}},} & \left( 73^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.

The criteria over the ROI is below-described. The processes 1 and 2(method 3-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (74) or (74′) with respect to threshold valuesTratioroi or Tdiffroi.

$\begin{matrix}{\frac{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}(x)}}^{2}}{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}^{2}} \leq {Tratioroi}} & (74) \\{or} & \; \\{{{\sum\limits_{x \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}}} \leq {Tdiffroi}},} & \left( 74^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.(Process 4: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component Distribution)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component distribution. The processes 1, 2 and 3of method 3-2 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (75) or (75′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}(x)}}^{2}}{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}^{2}} \leq {aboveTratioroi}}{or}} & (75) \\{{{\sum\limits_{x \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}}} \leq {aboveTdiffroi}},} & \left( 75^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.

Final estimate is obtained from eq. (71) or eq. (72).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 3-3]

The flowchart of the method 3-3 is shown in FIG. 12. To prevent phasematching from diverging at the process 1 of the method 3-1, estimationerror is reduced of the residual component. Possibility of divergence isdetected from above-described condition (70) or (70′), and byeffectively utilizing method 3-1 and 3-2, even if the SNR of theultrasound echo signals are low, accurate 1D displacement componentmeasurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 3-2 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

Phase matching and estimation of the 1D residual displacement componentare performed at every point x in the 1D ROI. That is, the processes 1and 2 of the method 3-1 are performed once at every point in the ROI.Thus, the estimate of the 1D residual component distribution isobtained.

During this estimation, if neither condition (70) nor (70′) is satisfiedwith, the method 3-1 is used. If condition (70) or (70′) is satisfiedwith at points or regions, in the process 2 of the method 3-2, oversufficiently large regions centered on the points or regions, or overthe ROI, the estimate dx^(i)(x) of the 1D displacement component dx(x)can be 1D low pass filtered or 1D median filtered as eq. (76).

_(x) ^(i)(x)=LPF[dx ^(i)(x)], or

_(x) ^(i)(x)=MED[dx ^(i)(x)]  (76)

Thus, the estimation error is reduced of the residual component comparedwith process 2 of the method 3-1 [eq. (63)].

Thus, iterative estimation is terminated at the process 5 of the method3-1 or the process 4 of the method 3-2. Hence, final estimate isobtained from eq. (67), or eq. (71), or eq. (76).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 3-4]

The flowchart of the method 3-4 is shown in FIG. 13. To prevent phasematching from diverging at the process 1 of the method 3-1, estimationerror is reduced of the residual component. Thus, even if the SNR of theultrasound echo signals are low, accurate 1D displacement componentmeasurement can be realized.

The procedure of iterative estimation is different from that of themethod 3-1. At i th estimate (i≧1), the following processes areperformed.

(Process 1: Estimation of the 1D Residual Displacement ComponentDistribution)

Phase matching and estimation of the 1D residual displacement componentare performed at every point x in the 1D ROI. That is, the process 1 ofthe method 3-1 is performed once at every point in the ROI.

To obtain the estimate

_(x) ^(i)(x) of the residual component distribution u^(i) _(x)(x), atevery point local 1D echo cross-spectrum is evaluated from the 1DFourier's transforms of the pre-deformation local 1D ultrasound echosignal r₁(l) and phase-matched post-deformation local 1D ultrasound echosignal r^(i) ₂(l). Alternatively, when pre-deformation local 1Dultrasound echo signal is phase-matched, at every point cross-spectrumof r^(i) ₁(l) and r₂(l) is evaluated.

The least squares method is implemented on the gradient of the phasewith utilization of each weight function, i.e., the squaredcross-spectrum |S_(2,1) ^(i)(l)|², where each weight function isnormalized by the power of the cross-spectrum, i.e.,

$\sum\limits_{l}{{{S_{2,1}^{i}(l)}}^{2}.}$Moreover, regularization method is also implemented. That is, byminimizing the next functional with respect to the vector u^(i)comprised of the 1D residual component distribution u_(x) ^(i)(x).

$\begin{matrix}{{{error}\left( u^{i} \right)} = {{{a - {Fu}^{i}}}^{2} + {\alpha_{1i}{{u^{i}{^{2}{{+ \alpha_{2i}}{{{Gu}^{i}{^{2}{{+ \alpha_{3i}}{{G^{T}{Gu}^{i}{^{2}{{+ \alpha_{4i}}{GG}^{T}{Gu}^{i}^{2}}}}}}}}}}}}}}}} & (77)\end{matrix}$

-   where a: vector comprised of x distribution of the cross-spectrum    phase Θ^(i)(l) weighted with cross-spectrum |S_(2,1) ^(i)(l)|    normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l}{{S_{2,1}^{i}(l)}}^{2}}$

-    evaluated at every point in the 1D ROI.    -   F: matrix comprised of x distribution of the Fourier's        coordinate value l weighted with cross-spectrum |S_(2,1)        ^(i)(l)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l}{{S_{2,1}^{i}(l)}}^{2}}$

-   -    evaluated at every point in the 1D ROI.    -   α_(1i), α_(2i), α_(3i), α_(4i): regularization parameter (at        least larger than zero)    -   Gu^(i): vector comprised of the finite difference approximations        of the 1D distribution of the 1D gradient components of the        unknown 1D residual components u_(x) ^(i)(x)

$\frac{\partial}{\partial x}{u_{x}^{i}(x)}$

-   -   G^(T)Gu^(i): vector comprised of the finite difference        approximations of the 1D distribution of the 1D Laplacians of        the unknown 1D residual components u_(x) ^(i)(x)

$\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}(x)}$

-   -   GG^(T)Gu^(i): vector comprised of the finite difference        approximations of the 1D distribution of the 1D gradient        components of the 1D Laplacians of the unknown 1D residual        components u_(x) ^(i)(x)

$\frac{\partial}{\partial x}\left( {\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}(x)}} \right)$

As ∥u^(i)∥², ∥Gu^(i)∥², ∥G^(T)Gu^(i)∥², ∥GG^(T)Gu^(i)∥² are positivedefinite, error(u^(i)) has one minimum value. Thus, by solving forresidual displacement component distribution u_(x) ^(i)(x) thesimultaneous equations:

$\begin{matrix}{{{\left( {{F^{T}F} + {\alpha_{1i}I} + {\alpha_{2i}G^{T}G} + {\alpha_{3i}G^{T}{GG}^{T}G} + {\alpha_{4i}G^{T}{GG}^{T}{GG}^{T}G}} \right)u^{i}} = {F^{T}a}},} & (78)\end{matrix}$estimate

_(x) ^(i)(x) of the residual component distribution u^(i) _(x)(x) isstably obtained. Thus, estimation error is reduced of the residualcomponent.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameters depend on the correlation ofthe local echo data (peak value of the cross-correlation function,sharpness of the cross-correlation function, width of thecross-correlation function), the SNR of the cross-spectrum power, etc.;then position of the unknown displacement component etc.

(Process 2: Update of the Estimate of the 1D Displacement ComponentDistribution)

The i−1 th estimate of the 1D displacement component distribution isupdated using i th estimate of the 1D residual component distribution.dx ^(i)(x)=

_(x) ^(i−1)(x)+

_(x) ^(i)(x )  (79)

Freely, this estimate can be 1D low pass filtered or 1D median filter toyield the estimate of the 1D displacement component distribution.

_(x) ^(i)(x)=LPF[dx ^(i)(x)], or

_(x) ^(i)(x)=MED [dx ^(i)(x)]  (80)Hence, phase matching of the process 1 of method 3-4 is performed usingthe 1D residual component data u_(x) ^(i)(x) obtained from eq. (78), orthe 1D component data dx^(i)(x) obtained from eq. (79), or smoothedestimate obtained from eq. (80).[Process 3: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement (Condition for Makingthe Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement, during iterative estimation, thelocal region used for each point is made small, or the local region usedover the ROI is made small.

The criteria for each point is below-described. The processes 1 and 2 ofmethod 3-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (81) or (81′) with respect to threshold valuesTratio or Tdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}_{x}^{i}(x)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}} \leq {Tratio}}{or}} & (81) \\{{{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}} \leq {Tdiff}},} & \left( 81^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.

The criteria over the ROI is below-described. The processes 1 and 2 ofmethod 3-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (82) or (82′) with respect to threshold valuesTratioroi or Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}(x)}}^{2}}{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}^{2}} \leq {Tratioroi}}{or}} & (82) \\{{{\sum\limits_{x \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}}} \leq {Tdiffroi}},} & \left( 82^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.(Process 4: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component Distribution)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component distribution. The processes 1, 2 and 3of method 3-4 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (83) or (83′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}(x)}}^{2}}{\sum\limits_{x \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}^{2}} \leq {aboveTratioroi}}{or}} & (83) \\{{{\sum\limits_{x \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}(x)} - {{\overset{\Cap}{u}}_{x}^{i - 1}(x)}}}} \leq {aboveTdiffroi}},} & \left( 83^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x)∥ and ∥

_(x) ^(i−1)(x)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual components.

Final estimate is obtained from eq. (79) or eq. (80).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 3-5]

The flowchart of the method 3-5 is shown in FIG. 14. To prevent phasematching from diverging at the process 1 of the method 3-1, estimationerror is reduced of the residual component. Possibility of divergence isdetected from above-described condition (70) or (70′), and byeffectively utilizing method 3-1 and 3-4, even if the SNR of theultrasound echo signals are low, accurate 1D displacement componentmeasurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 3-4 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

Phase matching and estimation of the 1D residual displacement componentare performed at every point x in the 1D ROI. That is, the process 1 ofthe method 3-1 is performed once at every point in the ROI. Moreover,using the regularization method, stably the estimate of the 1D residualcomponent distribution is obtained.

i−1 th estimate

_(x) ^(i−1)(x) of 1D displacement component distribution dx(x).

i th estimate

_(x) ^(i)(x) of 1D residual component distribution u_(x) ^(i)(x).

During this estimation, if neither condition (70) nor (70′) is satisfiedwith, the method 3-1 is used. If condition (70) or (70′) is satisfiedwith at points or regions, in the process 2 of the method 3-4, oversufficiently large regions centered on the points or regions, or overthe ROI, the estimate dx^(i)(x) of the 1D displacement component dx(x)can be 1D low pass filtered or 1D median filtered as eq. (84).

_(x) ^(i)(x)=LPF[dx ^(i)(x)], or

_(x) ^(i)(x)=MED[dx ^(i)(x)]  (84)Thus, the estimation error is reduced of the residual component.

Iterative estimation is terminated at the process 5 of the method 3-1 orthe process 4 of the method 3-4. Hence, final estimate is obtained fromeq. (67), or eq. (79), or eq. (84).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

(IV) Method 4: Measurement of 2D Displacement Component VectorDistribution in 3D SOI

[Method 4-1]

2D displacement component vector distribution in 3D SOI can be measuredby measuring 2D displacement component vector distribution in each (x,y)plane by means of the method 2-1, or 2-2, or 2-3, or 2-4, or 2-5 (FIG.21).

(Process 1)

In each (x,y) plane in 3D SOI, the method 2-1, or 2-2, or 2-3, or 2-4,or 2-5 is utilized. The initial estimate [eq. (29)] of the iterativeestimation of the 2D displacement vector in the 3D SOI is set as zerovector if a priori data is not given about displacements of body motionnor applied compression. Alternatively, values accurately estimated atneighborhood can be used (high correlation or least squares)

Moreover, the methods 4-2, 4-3, 4-4, and 4-5 are respectively baesd onthe methods 2-2, 2-3, 2-4, and 2-5.

[Method 4-2]

The flowchart of the method 4-2 is shown in FIG. 22. As example, let'sconsider measurement of 2D displacement vectord(x,y,z)[=(dx(x,y,z),dy(x,y,z))^(T)] in 3D SOI. At i th estimate (i≧1),the following processes are performed.

(Process 1: Estimation of the 2D Residual Displacement Component VectorDistribution in 3D SOI)

Phase matching and estimation of the 2D residual displacement vector areperformed at every point (x,y,z) in the 3D SOI. That is, the processes 1and 2 of the method 2-1 are performed once at every point in the 3D SOI.Thus, the i th estimate of the 2D residual component vector distributionu^(i)(x,y,z) is obtained as

^(i)(x,y,z)[=(

_(x) ^(i)(x,y,z),

_(y) ^(i)(x,y,z))^(T)].Process 2: Update of the Estimate of the 2D Displacement ComponentVector Distribution in 3D SOI)

The i−1 th estimate of the 2D displacement component vector distributionin the 3D SOI is updated using i th estimate of the 2D residualcomponent vector distribution in the 3D SOI.d ^(i)(x,y,z)=

^(i−1)(x,y,z)+

^(i)(x,y,z)  (85)

Next, this estimate is 3D low pass filtered or 3D median filter to yieldthe estimate of the 2D displacement component vector distribution:

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED[d ^(i)(x,y,z)].  (86)

Thus, the estimation error is reduced of the residual vector comparedwith process 2 of the method 2-1 [eq. (35)]. Hence, phase matching ofthe process 1 of method 4-2 is performed using smoothed estimate

^(i)(x,y,z) of the 2D displacement component vector distribution in the3D SOI.

[Process 3: Condition for Heightening the Spatial Resolution of the 2DDisplacement Component Vector Distribution Measurement in 3D SOI(Condition for Making the Local Region Small)]

In order to make the spatial resolution high of the 2D displacementcomponent vector distribution measurement in the 3D SOI, duringiterative estimation, the local region used for each point is madesmall, or the local region used over the 3D SOI is made small.

The criteria for each point is below-described. The processes 1 and 2(method 4-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (87) or (87′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (87) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 87^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (87) or (87′) can be applied to each direction component,and in this case the side is shorten in each direction.

The criteria over the 3D SOI is below-described. The processes 1 and 2(method 4-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (88) or (88′) with respect to threshold values Tratioroior Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {Tratioroi}}{or}} & (88) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {Tdiffroi}},} & \left( 88^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (88) or (88′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 4: Condition for Terminating the Iterative Estimation of the 2DDisplacement Component Vector Distribution in 3D SOI)

Below-described is the criteria for terminating the iterative estimationof the 2D displacement component vector distribution in the 3D SOI. Theprocesses 1, 2 and 3 of method 4-2 are iteratively carried out till thecriteria is satisfied with. For instance, the criteria is (89) or (89′)with respect to threshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (89) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {aboveTdiffroi}},} & \left( 89^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

Final estimate is obtained from eq. (85) or eq. (86).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 4-3]

The flowchart of the method 4-3 is shown in FIG. 23. As example, let'sconsider measurement of 2D displacement vectord(x,y,z)[=(dx(x,y,z),dy(x,y,z))^(T)] in 3D SOI.

Possibility of divergence is detected from above-described condition(42) or (42′) in above-described process 1 of the method 4-2, and byeffectively utilizing the method 4-1 based on the method 2-1, even ifthe SNR of the ultrasound echo signals are low, accurate 2D displacementvector measurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 4-2 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

The process 1 of the method 4-2 is performed. (Phase matching andestimation of the 2D residual displacement vector are performed at everypoint (x,y,z) in the 3D SOI.) That is, the processes 1 and 2 of themethod 2-1 are performed once at every point in the 3D SOI. Thus, theestimate of the 2D residual component vector distribution is obtained.

During this estimation, if neither condition (42) nor (42′) is satisfiedwith, the method 4-1 is used. If condition (42) or (42′) is satisfiedwith at points or regions, in the process 2 of the method 4-2, oversufficiently large regions centered on the points or regions, or overthe 3D SOI, the estimate d^(i)(x,y,z) [eq. (85)] of the 2D displacementvector d(x,y,z) can be 3D low pass filtered or 3D median filtered as eq.(90). Thus, the estimation error is reduced of the residual vectorcompared with process 2 of the method 2-1 [eq. (35)].

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED[d ^(i)(x,y,z)]  (90)

Thus, iterative estimation is terminated at the process 1 of the method4-1 based on the method 2-1, or the process 4 of the method 4-2. Hence,final estimate is obtained from eq. (39), or eq. (85), or eq. (90).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 4-4]

The flowchart of the method 4-4 is shown in FIG. 24. As example, let'sconsider measurement of 2D displacement vectord(x,y,z)[=(dx(x,y,z),dy(x,y,z))^(T)] in 3D SOI. At i th estimate (i≧1),the following process 1 is performed.

(Process 1: Estimation of the 2D Residual Displacement Component VectorDistribution in 3D SOI)

Phase matching and estimation of the 2D residual displacement vector areperformed at every point (x,y,z) in the 3D SOI. That is, the process 1of the method 2-1 is performed once at every point in the 3D SOI.

To obtain the estimate

^(i)(x,y,z)[=(

_(x) ^(i)(x,y,z),

_(y) ^(i)(x,y,z))^(T)] of the 2D residual component vector distributionu^(i)(x,y,z)[=(u^(i) _(x)(x,y,z), u^(i) _(y)(x,y,z))^(T)] in the 3D SOI,at every point local 2D echo cross-spectrum [eq. (31)] is evaluated fromthe 2D Fourier's transforms of the pre-deformation local 2D ultrasoundecho signal r₁(l,m) and phase-matched post-deformation local 2Dultrasound echo signal r^(i) ₂(l,m) Alternatively, when pre-deformationlocal 2D ultrasound echo signal is phase-matched, at every pointcross-spectrum of r^(i) ₁(l,m) and r₂(l,m) is evaluated.

The least squares method is implemented on the gradient of the phasewith utilization of each weight function, i.e., the squaredcross-spectrum |S_(2,1) ^(i)(l,m)|², where each weight function isnormalized by the power of the cross-spectrum, i.e.,

$\sum\limits_{l,m}{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}$[eq. (34)]. Moreover, regularization method is also implemented. Thatis, by minimizing the next functional with respect to the vector u^(i)comprised of the 2D residual component vector distribution u^(i)(x,y,z)in the 3D SOI.

$\begin{matrix}{{{error}\left( u^{i} \right)} = {{{a - {Fu}^{i}}}^{2} + {\alpha_{1i}{{u^{i}{^{2}{{+ \alpha_{2i}}{{{Gu}^{i}{^{2}{{+ \alpha_{3i}}{{G^{T}{Gu}^{i}{^{2}{{+ \alpha_{4i}}{GG}^{T}{Gu}^{i}^{2}}}}}}}}}}}}}}}} & (91)\end{matrix}$

-   where a: vector comprised of (x,y,z) distribution of the    cross-spectrum phase Θ^(i)(l,m) weighted with cross-spectrum    |S_(2,1) ^(i)(l,m)| normalized by the magnitude of the    cross-spectrum

$\sqrt{\;}{\sum\limits_{l,m}{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}}$

-    evaluated at every point in the 3D SOI.    -   F: matrix comprised of (x,y,z) distribution of the Fourier's        coordinate value (l,m) weighted with cross-spectrum |S_(2,1)        ^(i)(l,m)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l,m}{{S_{2,1}^{i}\left( {l,m} \right)}}^{2}}$

-   -    evaluated at every point in the 3D SOI.    -   α_(1i), α_(2i), α_(3i), α_(4i): regularization parameter (at        least larger than zero)    -   Gu^(i): vector comprised of the finite difference approximations        of the 3D distributions of the 3D gradient components of the        unknown 2D residual vector u^(i)(x,y,z) components

${\frac{\partial}{\partial x}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial y}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial z}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial x}{u_{y}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial y}{u_{y}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial z}{u_{y}^{i}\left( {x,y,z} \right)}}$

-   -   G^(T)Gu^(i): vector comprised of the finite difference        approximations of the 3D distributions of the 3D Laplacian of        the unknown 2D residual vector u^(i)(x,y,z) components

${\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}$${\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}$

-   -   GG^(T)Gu^(i): vector comprised of the finite difference        approximations of the 3D distributions of the 3D gradient        components of the 3D Laplacians of the unknown 2D residual        vector u^(i)(x,y,z) components

${\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial z}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial z}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{y}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{y}^{i}\left( {x,y,z} \right)}}} \right)}$

As ∥u^(i)∥², ∥Gu^(i)∥², ∥G^(T)Gu^(i)∥², ∥GG^(T)Gu^(i)∥² are positivedefinite, error(u^(i)) has one minimum value. Thus, by solving for 2Dresidual displacement component vector distribution u^(i)(x,y,z) in the3D SOI the simultaneous equations:(F ^(T) F+α _(1i) I+α _(2i) G ^(T) G+α _(3i) G ^(T) GG ^(T) G+α _(4i) G^(T) GG ^(T) GG ^(T) G)u ^(i) =F ^(T) a,  (92-1)estimate

^(i)(x,y,z)[=(

_(x) ^(i)(x,y,z),

_(y) ^(i)(x,y,z))^(T)] of the 2D residual component vector distributionu^(i)(x,y,z)[=(u^(i) _(x)(x,y,z), u^(i) _(y)(x,y,z))^(T)] is stablyobtained. Thus, estimation error is reduced of the residual vector.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameters depend on the correlation ofthe local echo data (peak value of the cross-correlation function,sharpness of the cross-correlation function, width of thecross-correlation function), the SNR of the cross-spectrum power, etc.;then position of the unknown displacement vector, direction of theunknown displacement component, direction of the partial derivative,etc.

(Process 2: Update of the Estimate of the 2D Displacement ComponentVector Distribution)

The i−1 th estimate of the 2D displacement component vector distributionis updated using i th estimate of the 2D residual component vectordistribution.d ^(i)(x,y,z)=

^(i−1)(x,y,z)+

^(i)(x,y,z)  (92-2)

Freely, this estimate can be 3D low pass filtered or 3D median filter toyield the estimate of the 2D displacement component vector distribution.

^(i)(x,y,z)=LPF[

^(i)(x,y,z)], or ^(i)(x,y,z)=MED[d ^(i)(x,y,z)]  (93)Hence, phase matching of the process 1 of method 4-4 is performed usingthe 2D residual vector data u^(i)(x,y,z) obtained from eq. (91), or the2D vector data d^(i)(x,y,z) obtained from eq. (92-2), or smoothedestimate obtained from eq. (93).[Process 3: Condition for Heightening the Spatial Resolution of the 2DDisplacement Component Vector Distribution Measurement in 3D SOI(Condition for Making the Local Region Small)]

In order to make the spatial resolution high of the 2D displacementcomponent vector distribution measurement, during iterative estimation,the local region used for each point is made small, or the local regionused over the 3D SOI is made small.

The criteria for each point is below-described. The processes 1 and 2 ofmethod 4-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (94) or (94′) with respect to threshold values Tratio orTdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (94) \\{{{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 94^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (94) or (94′) can be applied to each direction component,and in this case the side is shorten in each direction.

The criteria over the 3D SOI is below-described. The processes 1 and 2of method 4-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of each side is made half. For instance,the criteria is (95) or (95′) with respect to threshold values Tratioroior Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {Tratioroi}}{or}} & (95) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {Tdiffroi}},} & \left( 95^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

The condition (95) or (95′) can be applied to each direction component,and in this case the side is shorten in each direction.

(Process 4: Condition for Terminating the Iterative Estimation of the 2DDisplacement Component Vector Distribution in 3D SOI)

Below-described is the criteria for terminating the iterative estimationof the 2D displacement component vector distribution. The processes 1, 2and 3 of method 4-4 are iteratively carried out till the criteria issatisfied with. For instance, the criteria is (96) or (96′) with respectto threshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (96) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}^{i - 1}\left( {x,y,z} \right)}}}} \leq {aboveTdiffroi}},} & \left( 96^{\prime} \right)\end{matrix}$where ∥

^(i)(x,y,z)∥ and ∥

^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th and i−1th estimates of the residual vectors.

Final estimate is obtained from eq. (92-2) or eq. (93).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 4-5]

The flowchart of the method 4-5 is shown in FIG. 25. As example, let'sconsider measurement of 2D displacement vectord(x,y,z)[=(dx(x,y,z),dy(x,y,z))^(T)] in 3D SOI.

Possibility of divergence is detected from above-described condition(42) or (42′) in above-described process 1 of the method 4-4, and byeffectively utilizing the method 4-1 based on the method 2-1, even ifthe SNR of the ultrasound echo signals are low, accurate 2D displacementvector measurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 4-4 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

The process 1 of the method 4-4 is performed. (Phase matching andestimation of the 2D residual displacement vector are performed at everypoint (x,y,z) in the 3D SOI.) That is, the process 1 of the method 2-1is performed once at every point in the 3D SOI. Moreover, using theregularization method, stably the estimate of the 2D residual componentvector distribution is obtained.

i−1 th estimate

^(i−1)(x,y,z) of 2D displacement component vector distribution d(x,y,z).

i th estimate

^(i)(x,y,z) of 2D residual component vector distribution u^(i)(x,y,z).

During this estimation, if neither condition (42) nor (42′) is satisfiedwith, the method 4-1 based on the method 2-1 is used. If condition (42)or (42′) is satisfied with at points or regions, in the process 2 of themethod 4-4, over sufficiently large regions centered on the points orregions, or over the 3D SOI, the estimate d^(i)(x,y,z) of the 2Ddisplacement vector d(x,y,z) can be 3D low pass filtered or 3D medianfiltered as eq. (97).

^(i)(x,y,z)=LPF[d ^(i)(x,y,z)], or

^(i)(x,y,z)=MED[d ^(i)(x,y,z)]  (97)Thus, the estimation error is reduced of the residual vector.

Iterative estimation is terminated at the process 1 of the method 4-1based on the method 2-1, or the process 4 of the method 4-4. Hence,final estimate is obtained from eq. (39), or eq. (92-2), or eq. (97).

The initial estimate [eq. (29)] of the iterative estimation of the 2Ddisplacement vector is set as zero vector if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

(V) Method 5: Measurement of 1D Displacement (One Direction) ComponentDistribution in 3D SOI

[Method 5-1]

1D x displacement component distribution in 3D SOI can be measured bymeasuring 1D x displacement component distribution in each line beingparallel to x axis by means of the method 3-1, or 3-2, or 3-3, or 3-4,or 3-5 (FIG. 21).

(Process 1)

In each line being parallel to x axis in 3D SOI, the method 3-1, or 3-2,or 3-3, or 3-4, or 3-5 is utilized. The initial estimate [eq. (57)] ofthe iterative estimation of the 1D displacement component in the 3D SOIis set as zero if a priori data is not given about displacements of bodymotion nor applied compression. Alternatively, values accuratelyestimated at neighborhood can be used (high correlation or leastsquares).

Moreover, the methods 5-2, 5-3, 5-4, and 5-5 are respectively baesd onthe methods 3-2, 3-3, 3-4, and 3-5.

[Method 5-2]

The flowchart of the method 5-2 is shown in FIG. 22. As example, let'sconsider measurement of 1D displacement component dx(x,y,z) in 3D SOI.At i th estimate (i≧1), the following processes are performed.

(Process 1: Estimation of the 1D Residual Displacement ComponentDistribution in 3D SOI)

Phase matching and estimation of the 1D residual displacement componentare performed at every point (x,y,z) in the 3D SOI. That is, theprocesses 1 and 2 of the method 3-1 are performed once at every point inthe 3D SOI. Thus, the i th estimate of the 1D residual componentdistribution u_(x) ^(i)(x,y,z) is obtained as

_(x) ^(i)(x,y,z).

(Process 2: Update of the Estimate of the 1D Displacement ComponentDistribution in 3D SOI)

The i−1 th estimate of the 1D displacement component distribution in the3D SOI is updated using i th estimate of the 1D residual componentdistribution in the 3D SOI.dx ^(i)(x,y,z)=

_(x) ^(i−1)(x,y,z)+

_(x) ^(i)(x,y,z)  (98)

Next, this estimate is 3D low pass filtered or 3D median filter to yieldthe estimate of the 1D displacement component distribution:

_(x) ^(i)(x,y,z)=LPF[dx ^(i)(x,y,z)], or

_(x) ^(i)(x,y,z)=MED[dx ^(i)(x,y,z)]  (99)

Thus, the estimation error is reduced of the residual component comparedwith process 2 of the method 3-1 [eq. (63)]. Hence, phase matching ofthe process 1 of method 5-2 is performed using smoothed estimate

_(x) ^(i)(x,y,z) of the 1D displacement component distribution in the 3DSOI.

[Process 3: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement in 3D SOI (Condition forMaking the Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement in the 3D SOI, during iterativeestimation, the local region used for each point is made small, or thelocal region used over the 3D SOI is made small.

The criteria for each point is below-described. The processes 1 and 2(method 5-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (87) or (87′) with respect to threshold valuesTratio or Tdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (100) \\{{{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 100^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y,z)∥ and ∥

_(x) ^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

The criteria over the 3D SOI is below-described. The processes 1 and 2(method 5-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (101) or (101′) with respect to thresholdvalues Tratioroi or Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {Tratioroi}}{or}} & (101) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}}} \leq {Tdiffroi}},} & \left( 101^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y,z)∥ and ∥

_(x) ^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.(Process 4: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component Distribution in 3D SOI)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component distribution in the 3D SOI. Theprocesses 1, 2 and 3 of method 5-2 are iteratively carried out till thecriteria is satisfied with. For instance, the criteria is (102) or(102′) with respect to threshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (102) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}}} \leq {aboveTdiffroi}},} & \left( 102^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y,z)∥ and ∥

_(x) ^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

Final estimate is obtained from eq. (98) or eq. (99).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 5-3]

The flowchart of the method 5-3 is shown in FIG. 23. As example, let'sconsider measurement of 1D displacement component dx(x,y,z) in 3D SOI.

Possibility of divergence is detected from above-described condition(70) or (70′) in above-described process 1 of the method 5-2, and byeffectively utilizing the method 5-1 based on the method 3-1, even ifthe SNR of the ultrasound echo signals are low, accurate 1D displacementcomponent measurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 5-2 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

The process 1 of the method 5-2 is performed. (Phase matching andestimation of the 1D residual displacement component are performed atevery point (x,y, z) in the 3D SOI.) That is, the processes 1 and 2 ofthe method 3-1 are performed once at every point in the 3D SOI. Thus,the estimate of the 1D residual component distribution is obtained.

During this estimation, if neither condition (70) nor (70′) is satisfiedwith, the method 5-1 is used. If condition (70) or (70′) is satisfiedwith at points or regions, in the process 2 of the method 5-2, oversufficiently large regions centered on the points or regions, or overthe 3D SOI, the estimate dx^(i)(x,y,z) [eq. (98)] of the 1D displacementcomponent dx(x,y,z) can be 3D low pass filtered or 3D median filtered aseq. (102). Thus, the estimation error is reduced of the residualcomponent compared with process 2 of the method 3-1 [eq. (63)].

_(x) ^(i)(x,y,z)=LPF[dx ^(i)(x,y,z)], or

_(x) ^(i)(x,y,z)=MED[dx ^(i)(x,y,z)]  (102)

Thus, iterative estimation is terminated at the process 1 of the method5-1 based on the method 3-1, or the process 4 of the method 5-2. Hence,final estimate is obtained from eq. (67), or eq. (98), or eq. (102).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 5-4]

The flowchart of the method 5-4 is shown in FIG. 24. As example, let'sconsider measurement of 1D displacement component dx(x,y,z) in 3D SOI.At i th estimate (i≧1), the following process 1 is performed.

(Process 1: Estimation of the 1D Residual Displacement ComponentDistribution in 3D SOI)

Phase matching and estimation of the 1D residual displacement componentare performed at every point (x,y,z) in the 3D SOI. That is, the process1 of the method 3-1 is performed once at every point in the 3D SOI.

To obtain the estimate

_(x) ^(i)(x,y,z) of the 1D residual component distribution u^(i)_(x)(x,y,z) in the 3D SOI, at every point local 1D echo cross-spectrum[eq. (59)] is evaluated from the 1D Fourier's transforms of thepre-deformation local 1D ultrasound echo signal r₁(l) and phase-matchedpost-deformation local 1D ultrasound echo signal r^(i) ₂(l).Alternatively, when pre-deformation local 1D ultrasound echo signal isphase-matched, at every point cross-spectrum of r^(i) ₁(l) and r₂(l) isevaluated.

The least squares method is implemented on the gradient of the phasewith utilization of each weight function, i.e., the squaredcross-spectrum |S_(2,1) ^(i)(l)|², where each weight function isnormalized by the power of the cross-spectrum, i.e.,

$\sum\limits_{l}{{S_{2,1}^{i}(l)}}^{2}$[eq. (62)]. Moreover, regularization method is also implemented. Thatis, by minimizing the next functional with respect to the vector u^(i)comprised of the 1D residual component distribution u_(x) ^(i)(x,y,z) inthe 3D SOI.error(u ^(i))=∥a−Fu ^(i)∥²+α_(1i) ∥u ^(i)∥²+α_(2i) ∥Gu ^(i)∥²+α_(3i) ∥G^(T) Gu ^(i)∥²+α_(4i) ∥GG ^(T) Gu ^(i)∥²  (103)

-   where a: vector comprised of (x,y,z) distribution of the    cross-spectrum phase Θ^(i)(l) weighted with cross-spectrum |S_(2,1)    ^(i)(l)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l}{{S_{2,1}^{i}(l)}}^{2}}$

-    evaluated at every point in the 3D SOI.    -   F: matrix comprised of (x,y,z) distribution of the Fourier's        coordinate value l weighted with cross-spectrum |S_(2,1)        ^(i)(l)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l}{{S_{2,1}^{i}(l)}}^{2}}$

-   -    evaluated at every point in the 3D SOI.    -   α_(1i), α_(2i), α_(3i), α_(4i): regularization parameter (at        least larger than zero)    -   Gu^(i): vector comprised of the finite difference approximations        of the 3D distributions of the 3D gradient components of the        unknown 1D residual component u_(x) ^(i)(x,y,z)

${\frac{\partial}{\partial x}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial y}{u_{x}^{i}\left( {x,y,z} \right)}},{\frac{\partial}{\partial z}{u_{x}^{i}\left( {x,y,z} \right)}}$

-   -   G^(T)Gu^(i): vector comprised of the finite difference        approximations of the 3D distributions of the 3D Laplacian of        the unknown 1D residual component u^(i)(x,y,z)

${\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}$

-   -   GG^(T)Gu^(i): vector comprised of the finite difference        approximations of the 3D distributions of the 3D gradient        components of the 3D Laplacians of the unknown 1D residual        component u_(x) ^(i)(x,y,z)

${\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},{\frac{\partial}{\partial z}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y,z} \right)}} + {\frac{\partial^{2}}{\partial z^{2}}{u_{x}^{i}\left( {x,y,z} \right)}}} \right)},$

As ∥u^(i)∥², ∥Gu^(i)∥², ∥G^(T)Gu^(i)∥², ∥GG^(T)Gu^(i)∥² are positivedefinite, error(u^(i)) has one minimum value. Thus, by solving for 1Dresidual displacement component distribution u_(x) ^(i)(x,y,z) in the 3DSOI the simultaneous equations:(F ^(T) F+α _(1i) I+α _(2i) G ^(T) G+α _(3i) G ^(T) GG ^(T) G+α _(4i) G^(T) GG ^(T) GG ^(T) G)u ^(i) =F ^(T) a,  (104)estimate

_(x) ^(i)(x,y,z) of the 1D residual component distribution u^(i)_(x)(x,y,z) is stably obtained. Thus, estimation error is reduced of theresidual component.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameters depend on the correlation ofthe local echo data (peak value of the cross-correlation function,sharpness of the cross-correlation function, width of thecross-correlation function), the SNR of the cross-spectrum power, etc.;then position of the unknown displacement component etc.

(Process 2: Update of the Estimate of the 1D Displacement ComponentDistribution)

The i−1 th estimate of the 1D displacement component distribution isupdated using i th estimate of the 1D residual component distribution.dx ^(i)(x,y,z)=

_(x) ^(i−1)(x,y,z)+

_(x) ^(i)(x,y,z)  (105)

Freely, this estimate can be 3D low pass filtered or 3D median filter toyield the estimate of the 1D displacement component distribution.

_(x) ^(i)(x,y,z)=LPF[dx ^(i)(x,y,z)], or

_(x) ^(i)(x,y,z)=MED[dx ^(i)(x,y,z)]  (106)Hence, phase matching of the process 1 of method 5-4 is performed usingthe 1D residual component data u_(x) ^(i)(x,y,z) obtained from eq.(104), or the 1D component data dx^(i)(x,y,z) obtained from eq. (105),or smoothed estimate obtained from eq. (106).[Process 3: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement in 3D SOI (Condition forMaking the Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement, during iterative estimation, thelocal region used for each point is made small, or the local region usedover the 3D SOI is made small.

The criteria for each point is below-described. The processes 1 and 2 ofmethod 5-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (107) or (107′) with respect to thresholdvalues Tratio or Tdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}} \leq {Tratio}}{or}} & (107) \\{{{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}} \leq {Tdiff}},} & \left( 107^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y,z)∥ and ∥

_(x) ^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

The criteria over the 3D SOI is below-described. The processes 1 and 2of method 5-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (108) or (108′) with respect to thresholdvalues Tratioroi or Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {Tratioroi}}{or}} & (108) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}}} \leq {Tdiffroi}},} & \left( 108^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y,z)∥ and ∥

_(x) ^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.(Process 4: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component Distribution in 3D SOI)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component distribution. The processes 1, 2 and 3of method 5-4 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (109) or (109′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)}}^{2}}{\sum\limits_{{({x,y,z})} \in {SOI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (109) \\{{{\sum\limits_{{({x,y,z})} \in {SOI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y,z} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y,z} \right)}}}} \leq {aboveTdiffroi}},} & \left( 109^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y,z)∥ and ∥

_(x) ^(i−1)(x,y,z)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

Final estimate is obtained from eq. (105) or eq. (106).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 5-5]

The flowchart of the method 5-5 is shown in FIG. 25. As example, let'sconsider measurement of 1D displacement component dx(x,y,z) in 3D SOI.

Possibility of divergence is detected from above-described condition(70) or (70′) in above-described process 1 of the method 5-4, and byeffectively utilizing the method 5-1 based on the method 3-1, even ifthe SNR of the ultrasound echo signals are low, accurate 1D displacementcomponent measurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 5-4 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

The process 1 of the method 5-4 is performed. (Phase matching andestimation of the 1D residual displacement component are performed atevery point (x,y,z) in the 3D SOI.) That is, the process 1 of the method3-1 is performed once at every point in the 3D SOI. Moreover, using theregularization method, stably the estimate of the 1D residual componentdistribution is obtained.

i−1 th estimate

_(x) ^(i−1)(x,y,z) of 1D displacement component distribution dx(x,y,z).

i th estimate

_(x) ^(i)(x,y,z) of 1D residual component distribution u_(x)^(i)(x,y,z).

During this estimation, if neither condition (70) nor (70′) is satisfiedwith, the method 5-1 based on the method 3-1 is used. If condition (70)or (70′) is satisfied with at points or regions, in the process 2 of themethod 5-4, over sufficiently large regions centered on the points orregions, or over the 3D SOI, the estimate dx^(i)(x,y,z) of the 1Ddisplacement component dx(x,y,z) can be 3D low pass filtered or 3Dmedian filtered as eq. (110).

_(x) ^(i)(x,y,z)=LPF[dx ^(i)(x,y,z)], or

_(x) ^(i)(x,y,z)=MED[dx ^(i)(x,y,z)]  (110)Thus, the estimation error is reduced of the residual component.

Iterative estimation is terminated at the process 1 of the method 5-1based on the method 3-1, or the process 4 of the method 5-4. Hence,final estimate is obtained from eq. (67), or eq. (105), or eq. (110).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

(VI) Method 6: Measurement of 1D Displacement (One Direction) ComponentDistribution in 2D ROI

[Method 6-1]

1D x displacement component distribution in 2D ROI can be measured bymeasuring 1D x displacement component distribution in each line beingparallel to x axis by means of the method 3-1, or 3-2, or 3-3, or 3-4,or 3-5 (FIG. 21).

(Process 1)

In each line being parallel to x axis in 2D ROI, the method 3-1, or 3-2,or 3-3, or 3-4, or 3-5 is utilized. The initial estimate [eq. (57)] ofthe iterative estimation of the 1D displacement component in the 2D ROIis set as zero if a priori data is not given about displacements of bodymotion nor applied compression. Alternatively, values accuratelyestimated at neighborhood can be used (high correlation or leastsquares).

Moreover, the methods 6-2, 6-3, 6-4, and 6-5 are respectively baesd onthe methods 3-2, 3-3, 3-4, and 3-5.

[Method 6-2]

The flowchart of the method 6-2 is shown in FIG. 22. As example, let'sconsider measurement of 1D displacement component dx(x,y) in 2D ROI. Ati th estimate (i≧1), the following processes are performed.

(Process 1: Estimation of the 1D Residual Displacement ComponentDistribution in 2D ROI)

Phase matching and estimation of the 1D residual displacement componentare performed at every point (x,y) in the 2D ROI. That is, the processes1 and 2 of the method 3-1 are performed once at every point in the 2DROI. Thus, the i th estimate of the 1D residual component distributionu_(x) ^(i)(x,y) is obtained as

_(x) ^(i)(x,y).

(Process 2: Update of the Estimate of the 1D Displacement ComponentDistribution in 2D ROI)

The i−1 th estimate of the 1D displacement component distribution in the2D ROI is updated using i th estimate of the 1D residual componentdistribution in the 2D ROI.dx ^(i)(x,y)=

_(x) ^(i−1)(x,y)+

_(x) ^(i)(x,y)  (111)

Next, this estimate is 2D low pass filtered or 2D median filter to yieldthe estimate of the 1D displacement component distribution:

_(x) ^(i)(x,y)=LPF[dx ^(i)(x,y)], or

_(x) ^(i)(x,y)=MED[dx ^(i)(x,y)].  (112)

Thus, the estimation error is reduced of the residual component comparedwith process 2 of the method 3-1 [eq. (63)]. Hence, phase matching ofthe process 1 of method 6-2 is performed using smoothed estimate

_(x) ^(i)(x,y) of the 1D displacement component distribution in the 2DROI.

[Process 3: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement in 2D ROI (Condition forMaking the Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement in the 2D ROI, during iterativeestimation, the local region used for each point is made small, or thelocal region used over the 2D ROI is made small.

The criteria for each point is below-described. The processes 1 and 2(method 6-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (113) or (113′) with respect to thresholdvalues Tratio or Tdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}} \leq {Tratio}}{or}} & (113) \\{{{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}} \leq {Tdiff}},} & \left( 113^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y)∥ and ∥

_(x) ^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

The criteria over the 2D ROI is below-described. The processes 1 and 2(method 6-2) are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (114) or (114′) with respect to thresholdvalues Tratioroi or Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {Tratioroi}}{or}} & (114) \\{{{\sum\limits_{{({x,y})} \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}}} \leq {Tdiffroi}},} & \left( 114^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y)∥ and ∥

_(x) ^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.(Process 4: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component Distribution in 2D ROI)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component distribution in the 2D ROI. Theprocesses 1, 2 and 3 of method 6-2 are iteratively carried out till thecriteria is satisfied with. For instance, the criteria is (115) or(115′) with respect to threshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {aboveTratioroi}}{or}} & (115) \\{{{\sum\limits_{{({x,y})} \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}}} \leq {aboveTdiffroi}},} & \left( 115^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y)∥ and ∥

_(x) ^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

Final estimate is obtained from eq. (111) or eq. (112).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 6-3]

The flowchart of the method 6-3 is shown in FIG. 23. As example, let'sconsider measurement of 1D displacement component dx(x,y) in 2D ROI.

Possibility of divergence is detected from above-described condition(70) or (70′) in above-described process 1 of the method 6-2, and byeffectively utilizing the method 6-1 based on the method 3-1, even ifthe SNR of the ultrasound echo signals are low, accurate 1D displacementcomponent measurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 6-2 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

The process 1 of the method 6-2 is performed. (Phase matching andestimation of the 1D residual displacement component are performed atevery point (x,y) in the 2D ROI.) That is, the processes 1 and 2 of themethod 3-1 are performed once at every point in the 2D ROI. Thus, theestimate of the 1D residual component distribution is obtained.

During this estimation, if neither condition (70) nor (70′) is satisfiedwith, the method 6-1 is used. If condition (70) or (70′) is satisfiedwith at points or regions, in the process 2 of the method 6-2, oversufficiently large regions centered on the points or regions, or overthe 2D ROI, the estimate dx^(i)(x,y) [eq. (111)] of the 1D displacementcomponent dx(x,y) can be 2D low pass filtered or 2D median filtered aseq. (116). Thus, the estimation error is reduced of the residualcomponent compared with process 2 of the method 3-1 [eq. (63)].

_(x) ^(i)(x,y)=LPF[dx ^(i)(x,y)], or

_(x) ^(i)(x,y)=MED[dx ^(i)(x,y)]  (116)

Thus, iterative estimation is terminated at the process 1 of the method6-1 based on the method 3-1, or the process 4 of the method 6-2. Hence,final estimate is obtained from eq. (67), or eq. (111), or eq. (116).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 6-4]

The flowchart of the method 6-4 is shown in FIG. 24. As example, let'sconsider measurement of 1D displacement component dx(x,y) in 2D ROI. Ati th estimate (i≧1), the following process 1 is performed.

(Process 1: Estimation of the 1D Residual Displacement ComponentDistribution in 2D ROI)

Phase matching and estimation of the 1D residual displacement componentare performed at every point (x,y) in the 2D ROI. That is, the process 1of the method 3-1 is performed once at every point in the 2D ROI.

To obtain the estimate

_(x) ^(i)(x,y) of the 1D residual component distribution u^(i) _(x)(x,y)in the 2D ROI, at every point local 1D echo cross-spectrum [eq. (59)] isevaluated from the 1D Fourier's transforms of the pre-deformation local1D ultrasound echo signal r₁(1) and phase-matched post-deformation local1D ultrasound echo signal r^(i) ₂ (l) Alternatively, whenpre-deformation local 1D ultrasound echo signal is phase-matched, atevery point cross-spectrum of r^(i) ₁(l) and r₂(l) is evaluated.

The least squares method is implemented on the gradient of the phasewith utilization of each weight function, i.e., the squaredcross-spectrum |S_(2,1) ^(i)(l)|², where each weight function isnormalized by the power of the cross-spectrum, i.e.,

$\sum\limits_{l}{{S_{2,1}^{i}(l)}}^{2}$[eq. (62)]. Moreover, regularization method is also implemented. Thatis, by minimizing the next functional with respect to the vector u^(i)comprised of the 1D residual component distribution u_(x) ^(i)(x,y) inthe 2D ROI.

$\begin{matrix}{{{error}\left( u^{i} \right)} = {{{a - {Fu}^{i}}}^{2} + {\alpha_{1i}{u^{i}}^{2}} + {\alpha_{2i}{{Gu}^{i}}^{2}} + {\alpha_{3i}{{G^{T}{Gu}^{i}}}^{2}} + {\alpha_{4i}{{{GG}^{T}{Gu}^{i}}}^{2}}}} & (117)\end{matrix}$

-   where a: vector comprised of (x,y) distribution of the    cross-spectrum phase Θ^(i)(l) weighted with cross-spectrum |S_(2,1)    ^(i)(l)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{i}{\mspace{11mu}{S_{\;{2,\mspace{11mu} 1}}^{\; i}\;(l)}}^{2}}$

-    evaluated at every point in the 2D ROI.    -   F: matrix comprised of (x,y) distribution of the Fourier's        coordinate value l weighted with cross-spectrum |S_(2,1)        ^(i)(l)| normalized by the magnitude of the cross-spectrum

$\sqrt{\;}{\sum\limits_{l}{\;{S_{\;{2,\; 1}}^{\; i}(l)}}^{2}}$

-   -    evaluated at every point in the 2D ROI.    -   α_(1i), α_(2i), α_(3i), α_(4i): regularization parameter (at        least larger than zero)    -   Gu^(i): vector comprised of the finite difference approximations        of the 2D distributions of the 2D gradient components of the        unknown 1D residual component u_(x) ^(i)(x,y)

${\frac{\partial}{\partial x}{u_{x}^{i}\left( {x,y} \right)}},{\frac{\partial}{\partial y}{u_{x}^{i}\left( {x,y} \right)}}$

-   -   G^(T)Gu^(i): vector comprised of the finite difference        approximations of the 2D distributions of the 2D Laplacian of        the unknown 1D residual component u^(i)(x,y)

${\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y} \right)}}$

-   -   GG^(T)Gu^(i): vector comprised of the finite difference        approximations of the 2D distributions of the 2D gradient        components of the 2D Laplacians of the unknown 1D residual        component u_(x) ^(i)(x,y)

${\frac{\partial}{\partial x}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y} \right)}}} \right)},{\frac{\partial}{\partial y}\left( {{\frac{\partial^{2}}{\partial x^{2}}{u_{x}^{i}\left( {x,y} \right)}} + {\frac{\partial^{2}}{\partial y^{2}}{u_{x}^{i}\left( {x,y} \right)}}} \right)}$

As ∥u^(i)∥², ∥Gu^(i)∥², ∥G^(T)Gu^(i)∥², ∥GG^(T)Gu^(i)∥² are positivedefinite, error (u^(i)) has one minimum value. Thus, by solving for 1Dresidual displacement component distribution u_(x) ^(i)(x,y) in the 2DROI the simultaneous equations:

$\begin{matrix}{{{\left( {{F^{T}F} + {\alpha_{1i}I} + {\alpha_{2i}G^{T}G} + {\alpha_{3i}G^{T}{GG}^{T}G} + {\alpha_{4i}G^{T}{GG}^{T}{GG}^{T}G}} \right)u^{i}} = {F^{T}a}},} & (118)\end{matrix}$estimate

_(x) ^(i)(x,y) of the 1D residual component distribution u^(i) _(x)(x,y)is stably obtained. Thus, estimation error is reduced of the residualcomponent.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameters depend on the correlation ofthe local echo data (peak value of the cross-correlation function,sharpness of the cross-correlation function, width of thecross-correlation function), the SNR of the cross-spectrum power, etc.;then position of the unknown displacement component etc.

(Process 2: Update of the Estimate of the 1D Displacement ComponentDistribution)

The i−1 th estimate of the 1D displacement component distribution isupdated using i th estimate of the 1D residual component distribution.dx ^(i)(x,y)=

_(x) ^(i−1)(x,y)+

_(x) ^(i)(x,y)  (119)

Freely, this estimate can be 2D low pass filtered or 2D median filter toyield the estimate of the 1D displacement component distribution.

_(x) ^(i)(x,y)=LPF[dx ^(i)(x,y)], or

_(x) ^(i−1)(x,y)=MED[dx ^(i)(x,y)]  (120)Hence, phase matching of the process 1 of method 6-4 is performed usingthe 1D residual component data u_(x) ^(i)(x,y) obtained from eq. (118),or the 1D component data dx^(i)(x,y) obtained from eq. (119), orsmoothed estimate obtained from eq. (120).[Process 3: Condition for Heightening the Spatial Resolution of the 1DDisplacement Component Distribution Measurement in 2D ROI (Condition forMaking the Local Region Small)]

In order to make the spatial resolution high of the 1D displacementcomponent distribution measurement, during iterative estimation, thelocal region used for each point is made small, or the local region usedover the 2D ROI is made small.

The criteria for each point is below-described. The processes 1 and 2 ofmethod 6-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (121) or (121′) with respect to thresholdvalues Tratio or Tdiff.

$\begin{matrix}{{\frac{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}} \leq {Tratio}}{or}} & (121) \\{{{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}} \leq {Tdiff}},} & \left( 121^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y)∥ and ∥

_(x) ^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

The criteria over the 2D ROI is below-described. The processes 1 and 2of method 6-4 are iteratively carried out till the criteria is satisfiedwith. When the criteria is satisfied with, the local region is madesmall, for instance, the length of the local region is made half. Forinstance, the criteria is (122) or (122′) with respect to thresholdvalues Tratioroi or Tdiffroi.

$\begin{matrix}{{\frac{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {Tratioroi}}{or}} & (122) \\{{{\sum\limits_{{({x,y})} \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}}} \leq {Tdiffroi}},} & \left( 122^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y)∥ and ∥

_(x) ^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.(Process 4: Condition for Terminating the Iterative Estimation of the 1DDisplacement Component Distribution in 2D ROI)

Below-described is the criteria for terminating the iterative estimationof the 1D displacement component distribution. The processes 1, 2 and 3of method 6-4 are iteratively carried out till the criteria is satisfiedwith. For instance, the criteria is (123) or (123′) with respect tothreshold values aboveTratioroi or aboveTdiffroi.

$\begin{matrix}{\frac{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)}}^{2}}{\sum\limits_{{({x,y})} \in {ROI}}{{{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}^{2}} \leq {aboveTratioroi}} & (123) \\{or} & \; \\{{{\sum\limits_{{({x,y})} \in {ROI}}{{{{\overset{\Cap}{u}}_{x}^{i}\left( {x,y} \right)} - {{\overset{\Cap}{u}}_{x}^{i - 1}\left( {x,y} \right)}}}} \leq {aboveTdiffroi}},} & \left( 123^{\prime} \right)\end{matrix}$where ∥

_(x) ^(i)(x,y)∥ and ∥

_(x) ^(i−1)(x,y)∥ are respectively norms (magnitudes) of the i th andi−1 th estimates of the residual components.

Final estimate is obtained from eq. (119) or eq. (120).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

[Method 6-5]

The flowchart of the method 6-5 is shown in FIG. 25. As example, let'sconsider measurement of 1D displacement component dx(x,y) in 2D ROI.

Possibility of divergence is detected from above-described condition(70) or (70′) in above-described process 1 of the method 6-4, and byeffectively utilizing the method 6-1 based on the method 3-1, even ifthe SNR of the ultrasound echo signals are low, accurate 1D displacementcomponent measurement can be realized.

At first, the procedure of iterative estimation is same as that of themethod 6-4 (Processes 1, 2, 3, and 4). At i th estimate, the followingprocesses are performed.

The process 1 of the method 6-4 is performed. (Phase matching andestimation of the 1D residual displacement component are performed atevery point (x,y) in the 2D ROI.) That is, the process 1 of the method3-1 is performed once at every point in the 2D ROI. Moreover, using theregularization method, stably the estimate of the 1D residual componentdistribution is obtained.

i−1 th estimate

_(x) ^(i−1)(x,y) of 1D displacement component distribution dx(x,y).

i th estimate

_(x) ^(i)(x,y) of 1D residual component distribution u_(x) ^(i)(x,y).

During this estimation, if neither condition (70) nor (70′) is satisfiedwith, the method 6-1 based on the method 3-1 is used. If condition (70)or (70′) is satisfied with at points or regions, in the process 2 of themethod 6-4, over sufficiently large regions centered on the points orregions, or over the 2D ROI, the estimate dx^(i)(x,y) of the 1Ddisplacement component dx(x,y) can be 2D low pass filtered or 2D medianfiltered as eq. (124).

_(x) ^(i)(x,y)=LPF[dx ^(i)(x,y)], or

_(x) ^(i)(x,y)=MED[dx ^(i)(x,y)]  (124)Thus, the estimation error is reduced of the residual component.

Iterative estimation is terminated at the process 1 of the method 6-1based on the method 3-1, or the process 4 of the method 6-4. Hence,final estimate is obtained from eq. (67), or eq. (119), or eq. (124).

The initial estimate [eq. (57)] of the iterative estimation of the 1Ddisplacement component is set as zero if a priori data is not givenabout displacements of body motion nor applied compression.Alternatively, values accurately estimated at neighborhood can be used(high correlation or least squares).

In the 3D SOI, the 3D displacement vector distribution can also bemeasured using method 4 or method 5 by changing the adapted direction.In the 2D ROI, the 2D displacement vector distribution can also bemeasured using method 6 by changing the adapted direction. Except forthreshold value for terminating the iterative estimation, otherthreshold values can be updated. Estimate can also be performednon-iteratively.

When applying the regularization method, in addition to the magnitude ofthe unknown displacement vector, spatial continuity anddifferentiability of the unknown displacement vector distribution,mechanical properties of tissue (e.g., incompressibility), andcompatibility conditions of displacement vector distribution anddisplacement component distribution, as the a priori knowledge, used istemporal continuity and differentiability of the unknown series of thedisplacement vector distribution and displacement componentdistribution. The regularization parameter depends on time-spacedimension number, direction of the unknown displacement component,position of the unknown displacement vector, time, etc.

Thus, as the displacement vector can be measured accurately,consequently, in addition to 3D strain tensor, accurately measured canbe 2D strain tensor, one strain component, 3D strain rate tensor, 2Dstrain rate tensor, one strain rate component, acceleration vector,velocity vector, etc.

(VII) Differential Filter

The strain tensor components can be obtained by spatial differentialfiltering with suitable cut off frequency in time domain or frequencydomain the measured 3D, or 2D displacement vector components, ormeasured 1D direction displacement component in the 3D, 2D, or 1D ROI.The strain rate tensor components, acceleration vector components, orvelocity vector components can be obtained by time differentialfiltering with suitable cut off frequency in time domain or frequencydomain the measured time series of displacement components, or straincomponents. The strain rate tensor components can be obtained from thestrain tensor components directly measured by below-described signalprocessing.

As above described, when measuring displacement from the gradient of theecho cross-spectrum phase, to result the more accurate measurementaccuracy, the least squares method can be applied with utilization asthe weight function of the squares of the cross-spectrum usuallynormalized by the cross-spectrum power, where, to stabilize themeasurement, the regularization method can be applied, by which a prioriinformation can be incorporated, i.e., about within the ROI themagnitude of the unknown displacement vector, spatial continuity anddifferentiability of the unknown displacement vector distribution etc.

Next, as the estimation methods of the displacements during theiterative estimation to update 3D, 2D, 1D displacement component, inorder to reduce calculation amount and shorten calculation time, othermethods are also described. These estimation methods can be used incombination, or one of them can be used. To realize real-timemeasurement, the estimate can also be performed non-iteratively.

In order to reduce calculation amount and shorten calculation time,calculation process is simplified. That is, as the cross-spectrum phaseθ(ωx, ωy, ωz) is represented as θ₂(ωx, ωy, ωz)−θ₁(ωx, ωy, ωz) using thephases θ₁(ωx, ωy, ωz) and θ₂(ωx, ωy, ωz) respectively obtained from 3DFourier's transforms R₁(ωx, ωy, ωz) and R₂(ωx, ωy, ωz) of the local echosignals under pre- and post-deformation, the displacement vectoru(=(ux,uy,uz)^(T)) is represented as

$\begin{matrix}{\begin{pmatrix}{ux} \\{uy} \\{uz}\end{pmatrix} = \begin{matrix}{{grad}\mspace{11mu}\left( {\arg\mspace{11mu}\left\lbrack {{R_{2}^{*}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}\mspace{11mu}{R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}} \right\rbrack} \right)} \\\left( {{{where}\mspace{14mu}{grad}} = \left( {{{\mathbb{d}\text{/}}{\mathbb{d}\omega}\; x},{{\mathbb{d}\text{/}}{\mathbb{d}\omega}\; y},{{\mathbb{d}\text{/}}{\mathbb{d}\omega}\; z}} \right)^{T}} \right)\end{matrix}} \\{= \begin{pmatrix}{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; x}\theta\mspace{11mu}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; y}\theta\mspace{11mu}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; z}\theta\mspace{11mu}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}\end{pmatrix}} \\{= \begin{pmatrix}{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; x}\left( {{\theta_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} - {\theta_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}} \right)} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; y}\left( {{\theta_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} - {\theta_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}} \right)} \\{\frac{\mathbb{d}\;}{{\mathbb{d}\omega}\; z}\left( {{\theta_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} - {\theta_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}} \right)}\end{pmatrix}} \\{= {{Im}\mspace{11mu}\left\lbrack {{grad}\mspace{11mu}\left( {\ln\mspace{11mu}\left\{ {{R_{2}^{*}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}\mspace{11mu}{R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}} \right\}} \right)} \right\rbrack}} \\{= \begin{pmatrix}\begin{matrix}{- \frac{\begin{matrix}{{{{Re}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\frac{\mathbb{d}}{{\mathbb{d}\omega}\; x}{{Im}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}} -} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; x}{{Re}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\mspace{14mu}{{Im}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}}\end{matrix}}{{{R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}}^{2}}} \\{+ \frac{\begin{matrix}{{{{Re}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\frac{\mathbb{d}}{{\mathbb{d}\omega}\; x}{{Im}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}} -} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; x}{{Re}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\mspace{11mu}{{Im}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}}\end{matrix}}{{{R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}}^{2}}}\end{matrix} \\{- \frac{\begin{matrix}{{{{Re}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\frac{\mathbb{d}}{{\mathbb{d}\omega}\; y}{{Im}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}} -} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; y}{{Re}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\mspace{11mu}{{Im}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}}\end{matrix}}{{{R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}}^{2}}} \\{+ \frac{\begin{matrix}{{{{Re}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\frac{\mathbb{d}}{{\mathbb{d}\omega}\; y}{{Im}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}} -} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; y}{{Re}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\mspace{11mu}{{Im}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}}\end{matrix}}{{{R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}}^{2}}} \\{- \frac{\begin{matrix}{{{{Re}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\frac{\mathbb{d}}{{\mathbb{d}\omega}\; z}{{Im}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}} -} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; z}{{Re}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\mspace{11mu}{{Im}\mspace{11mu}\left\lbrack {R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}}\end{matrix}}{{{R_{2}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}}^{2}}} \\{+ \frac{\begin{matrix}{{{{Re}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\frac{\mathbb{d}}{{\mathbb{d}\omega}\; z}{{Im}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}} -} \\{\frac{\mathbb{d}}{{\mathbb{d}\omega}\; z}{{Re}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}\mspace{11mu}{{Im}\mspace{11mu}\left\lbrack {R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)} \right\rbrack}}\end{matrix}}{{{R_{1}\left( {{\omega\; x},{\omega\; y},{\omega\; z}} \right)}}^{2}}}\end{pmatrix}}\end{matrix}$

Then, using the phases of the high SNR frequency the displacement vectoru can be obtained by partial differentiating in the frequency directionsωx, ωy, ωz the difference between the phases θ₂(ωx, ωy, ωz) and θ₁(ωx,ωy, ωz), or by calculating the difference between partialdifferentiations in the frequency directions ωx, ωy, ωz of the phasesθ₂(ωx, ωy, ωz) and θ₁(ωx, ωy, ωz), or by using Fourier's transformvalues Re[R₂(ωx, ωy, ωz)], Im[R₂(ωx, ωy, ωz)], Re[R₁(ωx, ωy, ωz)],Im[R₁(ωx, ωy, ωz)], and their partial derivatives in the frequencydirections ωx, ωy, ωz without unwrapping the phase. These partialderivatives can be obtained by finite-difference approximating ordifferential filtering. Freely, the phases, the signal components, ornumerator and denominator can be moving averaged or low-pass filtered inthe frequency domain. The final estimate can be the mean vectorcalculated from the displacement data obtained at high SNR frequencies.

The 2D displacement vector and one direction displacement component canrespectively be obtained in a similar way by calculating 2D and 1DFourier's transforms.

The simultaneous equations of the above-described equation can be solvedin the frequency domain, or spatial and temporal simultaneous equationsof the above-described equation can be handled, where above-describedregularization method can be applied.

When performing 1D (one direction) calculation, in order to reducecalculation amount and shorten calculation time, calculation process issimplified. That is, for instance, when performing x directioncalculation, as the cross-spectrum phase θ(ωx, ωy, ωz) is represented asθ(ωx)=ux·ωx, the displacement is obtained form the phase of the high SNRfrequency (y direction calculation; θ(ωy)=uy·ωy, z directioncalculation; θ(ωz)=uz·ωz). The final estimate can be the mean valuecalculated from the displacement data obtained at high SNR frequencies.

When large displacement needs to be handled, before estimating thegradient of the cross-spectrum phase, the phase was unwrapped, or thedisplacement was coarsely estimated by cross-correlation method. Thus,measurement procedure had become complex one. Otherwise, to cope withthese complexity, the measurement procedure is made simpler withoutthese processes by introducing process of thinning out data and remakingdata interval original.

The simultaneous equations of the above-described equation can be solvedin the frequency domain, or spatial and temporal simultaneous equationsof the above-described equation can be handled, where above-describedregularization method can be applied.

Otherwise, echo signals are acquired at two different time, freely, theaouto-correlation method (beam direction or scan direction) and theregularization method can be equipped.

Otherwise, freely, ultrasound Doppler's method can be equipped. TheDoppler's shift can be detected in beam direction or scan direction.

There are many methods for detecting the Doppler's shift. From the phasedistributionθ_(ZR)(x,y,z,t)=tan⁻¹(Im[Z_(R)(x,y,z,t)]/Re[Z_(R)(x,y,z,t)]) of thequadrate demodulation signalZ_(R)(x,y,z,t)(=Re[Z_(R)(x,y,z,t)]+jIm[Z_(R)(x,y,z,t)] in R axisdirection acquired at each position (x,y,z) in the ROI, for instance,the velocity component vx in x axis direction (R=x) at time t=T and atposition (X,Y,Z) can be obtained as

$\begin{matrix}{{vx} = \left. {{- \frac{1}{s_{x}\pi}}{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {\frac{c_{x}}{f_{0x}}{\tan^{- 1}\left( \frac{{Im}\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack}{{Re}\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack} \right)}} \right\rbrack}} \right|_{{x = X},{y = Y},{z = Z},{t = T}}} \\{= \left. {{- \frac{1}{s_{x}\pi}}\left( {\frac{c_{x}}{f_{0x}}\frac{\begin{matrix}\begin{matrix}{{{Re}\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{Im}\left\lbrack {{Zx}\left( {x,,y,z,t} \right)} \right\rbrack} \right\rbrack} -}\end{matrix} \\\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{Re}\mspace{11mu}\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack} \right\rbrack} \cdot} \\{{Im}\;\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack}\end{matrix}\end{matrix}}{\begin{matrix}{{{Re}\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack}^{2} +} \\{{Im}\left\lbrack {{Zx}\left( {x,y,z,t} \right)} \right\rbrack}^{2}\end{matrix}}} \right)} \right|_{{x = X},{y = Y},{z = Z},{t = T}}}\end{matrix}$c_(R) is the ultrasound propagating velocity and scan velocityrespectively when R axis is beam axis and scan axis. f_(0R) isultrasound carrier frequency and sine frequency respectively when R axisis beam axis and scan axis. s_(R) is 4.0 and 2.0 respectively when Raxis is beam axis and scan axis. As above-described, temporal gradientof the phase θ_(ZR)(x,y,z,t) can also be obtained by finite differenceapproximating or differential filtering after obtaining the phaseθ_(ZR)(x,y,z,t). Freely, the phases, the signal components, or numeratorand denominator can be moving averaged or low-pass filtered in the timedomain. Thus, the velocity component distributions (series) can beobtained in the ROI.

The spatial and temporal simultaneous equations of the above-describedequation can be handled, where above-described regularization method canbe applied.

By multiplying pulse transmitting interval Ts to each velocity componentdistributions (series), the displacement component distribution (series)can be obtained. Alternatively, by integrating the velocity vectorcomponent distributions (series), the displacement vector distribution(series) can be obtained.

From temporal spatial derivatives of these velocity vector componentdistributions (series) or displacement vector component distributions(series), obtained are strain tensor component distributions (series),acceleration vector component distributions (series), and strain ratetensor component distributions (series).

Otherwise, freely, a method for directly obtaining strain tensorcomponents can be equipped, i.e., from spatial partial derivative of thephase of the quadrate demodulate signal (beam direction or scandirection) of the ultrasound echo signals.

From the phase distributionθ_(ZR)(x,y,z,t)=tan⁻¹(Im[Z_(R)(x,y,z,t)]/Re[Z_(R)(x,y,z,t)]) of thequadrate demodulation signalZ_(R)(x,y,z,t)(=Re[Z_(R)(x,y,z,t)]+jIm[Z_(R)(x,y,z,t)] in R axisdirection acquired at each position (x,y,z) in the ROI, for instance,the normal strain component εxx in x axis direction (R=x) at time t=Tand at position (X,Y,Z) can be obtained as

$\begin{matrix}{\begin{matrix}{{ɛ\;{xx}}\mspace{11mu}} \\\left( {X,Y,Z,T} \right)\end{matrix} = \left. {\frac{\partial}{\partial x}{u_{x}\left( {x,y,z,t} \right)}} \right|_{{x = X},{y = Y},{z = Z},{t = T}}} \\{= \left. {{- \frac{1}{s_{x}\pi}}\frac{\mathbb{d}}{\mathbb{d}x}{\frac{\mathbb{d}}{\mathbb{d}t}\begin{bmatrix}{\frac{c_{x}}{f_{0x}}\tan^{- 1}} \\\left( \frac{\underset{\_}{Im}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}{\underset{\_}{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right)\end{bmatrix}}} \middle| {}_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}{Ts} \right.} \\{= \left. {{- \frac{1}{s_{x}\pi}}\frac{\mathbb{d}}{\mathbb{d}t}\left( {\frac{c_{x}}{f_{0_{x}}}\frac{\begin{matrix}{{{Re}\mspace{11mu}\begin{bmatrix}{Zx} \\\left( {x,y,z,t} \right)\end{bmatrix}} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}x}\left\lbrack {{Im}\mspace{11mu}\begin{bmatrix}{Zx} \\{\mspace{11mu}\left( {x,y,z,t} \right)}\end{bmatrix}} \right\rbrack} -} \\{{\frac{\mathbb{d}}{\mathbb{d}x}\left\lbrack {{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right\rbrack} \cdot} \\{{Im}\mspace{11mu}\begin{bmatrix}{Zx} \\\left( {x,y,z,t} \right)\end{bmatrix}}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2} +} \\{{Im}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2}\end{matrix}}} \right)} \middle| {}_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}{Ts} \right.} \\{= \left. {{- \frac{1}{s_{x}\pi}}\frac{\mathbb{d}}{\mathbb{d}x}\left( {\frac{c_{x}}{f_{0_{x}}}\frac{\begin{matrix}\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{Im}\mspace{11mu}\begin{bmatrix}{Zx} \\{\mspace{11mu}\left( {x,y,z,t} \right)}\end{bmatrix}} \right\rbrack} -}\end{matrix} \\\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right\rbrack} \cdot} \\{{Im}\mspace{11mu}\begin{bmatrix}{Zx} \\\left( {x,y,z,t} \right)\end{bmatrix}}\end{matrix}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2} +} \\{{Im}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2}\end{matrix}}} \right)} \middle| {}_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}{Ts} \right.}\end{matrix}$

c_(R) is the ultrasound propagating velocity and scan velocityrespectively when R axis is beam axis and scan axis. f_(0R) isultrasound carrier frequency and sine frequency respectively when R axisis beam axis and scan axis. s_(R) is 4.0 and 2.0 respectively when Raxis is beam axis and scan axis. As above-described, spatial gradient ofthe phase θ_(ZR)(x,y,z,t) can also be obtained by finite differenceapproximating or differential filtering after obtaining the phaseθ_(ZR)(x,y,z,t). Freely, the phases, the signal components, or numeratorand denominator can be moving averaged or low-pass filtered in the spacedomain. For instance, the shear strain component εxy in x-y plane (R=xand y) at time t=T and at position (X,Y,Z) can be obtained as

$\begin{matrix}{\begin{matrix}{ɛ\;{xy}} \\\left( {X,Y,Z,T} \right)\end{matrix} = \left. {\frac{1}{2}\left( {{\frac{\partial}{\partial x}{uy}\mspace{11mu}\left( {x,y,z,t} \right)} + {\frac{\partial}{\partial y}{ux}\mspace{11mu}\left( {x,y,z,t} \right)}} \right)} \right|_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}} \\{= \left. {\frac{1}{2}\begin{pmatrix}{{- \frac{1}{s_{y}\pi}}\frac{\mathbb{d}}{\mathbb{d}x}\frac{\mathbb{d}}{\mathbb{d}t}} \\{\begin{bmatrix}{\frac{c_{y}}{f_{0_{y}}}\tan^{- 1}} \\\left( \frac{{Im}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}{{Re}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right)\end{bmatrix} -} \\{\frac{1}{s_{x}\pi}\frac{\mathbb{d}}{\mathbb{d}y}\frac{\mathbb{d}}{\mathbb{d}t}} \\\begin{bmatrix}{\frac{c_{x}}{f_{0_{x}}}\tan^{- 1}} \\\left( \frac{{Im}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}{{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right)\end{bmatrix}\end{pmatrix}} \middle| {}_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}{Ts} \right.} \\{= {\frac{1}{2}\left( {{{- \frac{1}{s_{y}\pi}}\frac{\mathbb{d}}{\mathbb{d}t}\left( {\frac{c_{y}}{f_{0_{y}}}\frac{\begin{matrix}\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \cdot \frac{\mathbb{d}}{\mathbb{d}x}} \\{\left\lbrack {{Im}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,,y,z,t} \right)} \right\rbrack} \right\rbrack -}\end{matrix} \\{{\frac{\mathbb{d}}{\mathbb{d}y}\begin{bmatrix}{Re} \\\left\lbrack {{Zy}\left( {x,y,z,t} \right)} \right\rbrack\end{bmatrix}} \cdot} \\{{Im}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zy}\mspace{14mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2} +} \\{{Im}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2}\end{matrix}}} \right)} -} \right.}} \\{\left. {\frac{1}{s_{x}\pi}\frac{\mathbb{d}}{\mathbb{d}t}\left( {\frac{c_{x}}{f_{0_{x}}}\frac{\begin{matrix}\begin{matrix}\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}y}\begin{bmatrix}{Im} \\{\;\left\lbrack {{Zx}\mspace{11mu}\left( {x,,y,z,t} \right)} \right\rbrack}\end{bmatrix}} -}\end{matrix} \\{{\frac{\mathbb{d}}{\mathbb{d}y}\begin{bmatrix}{Re} \\\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack\end{bmatrix}} \cdot}\end{matrix} \\{{Im}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right.}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2} +} \\{{Im}\mspace{11mu}\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2}\end{matrix}}} \right)} \right)_{\;_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}}{Ts}} \\{= {\frac{1}{2}\left( {{{- \frac{1}{s_{y}\pi}}\frac{\mathbb{d}}{\mathbb{d}x}\left( {\frac{c_{y}}{f_{0_{y}}}\frac{\begin{matrix}\begin{matrix}\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{Im}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right\rbrack} -}\end{matrix} \\{{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{Re}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right\rbrack} \cdot}\end{matrix} \\{{Im}\mspace{11mu}\left\lbrack {{Zy}\mspace{11mu}\left( {x,y,z,t} \right)} \right.}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\begin{bmatrix}{Zy} \\\left( {x,y,z,t} \right)\end{bmatrix}}^{2} +} \\{{Im}\mspace{11mu}\begin{bmatrix}{Zy} \\\left( {x,y,z,t} \right)\end{bmatrix}}^{2}\end{matrix}}} \right)} -} \right.}} \\{{\left. {\frac{1}{s_{x}\pi}\frac{\mathbb{d}}{\mathbb{d}y}\left( {\frac{c_{x}}{f_{0_{x}}}\frac{\begin{matrix}\begin{matrix}\begin{matrix}{{{Re}\mspace{11mu}\begin{bmatrix}{Zx} \\\left( {x,y,z,t} \right)\end{bmatrix}} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}t}\begin{bmatrix}{Im} \\{\;\left\lbrack {{Zx}\mspace{11mu}\left( {x,,y,z,t} \right)} \right\rbrack}\end{bmatrix}} -}\end{matrix} \\{{\frac{\mathbb{d}}{\mathbb{d}t}\begin{bmatrix}{Re} \\\left\lbrack {{Zx}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack\end{bmatrix}} \cdot}\end{matrix} \\{{Im}\mspace{11mu}\begin{bmatrix}{{Zx}\;} \\\left( {x,y,z,t} \right)\end{bmatrix}}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\begin{bmatrix}{Zx} \\\left( {x,y,z,t} \right)\end{bmatrix}}^{2} +} \\{{Im}\mspace{11mu}\begin{bmatrix}{Zx} \\\left( {x,y,z,t} \right)\end{bmatrix}}^{2}\end{matrix}}} \right)} \right)_{\;}}_{\begin{matrix}{{x = X},{y = Y},} \\{{z = Z},{t = T}}\end{matrix}}{Ts}}\end{matrix}$Thus, the strain component distributions (series) can be obtained in theROI.

The spatial and temporal simultaneous equations of the above-describedequation can be handled, where above-described regularization method canbe applied.

By integrating the partial derivatives of displacement vector componentdistributions (series), the displacement vector distribution (series)can be obtained.

From temporal spatial derivatives of these strain tensor componentdistributions (series) or displacement vector component distributions(series), obtained are strain rate tensor component distributions(series), and acceleration vector component distributions (series).

Freely, (I-1) complex cross-correlation method (phase in beam directionor scan direction of complex cross-correlation function signal obtainedfrom complex analytic signals or quadrate detection signals, or obtainedfrom cross-correlation of ultrasound echo signals) is utilized, or (I-2)both of complex cross-correlation method (beam direction or scandirection) and the regularization method are utilized, or (I-3) at least2D distribution (including beam direction or not) of the phase of 3D, or2D complex cross-correlation function signals or 1D complexcross-correlation function signal respectively obtained from 3D complexsignals with single-octant spectra, 2D complex signals withsingle-quadrant spectra, and conventional 1D complex analytic signal (S.L. Hahn, “Multidimensional complex signals with single-orthant spectra,”Proceedings of the IEEE, vol. 80, no. 8, pp. 1287-1300, 1992, where the3D and 2D complex signals are not proven to be analytic in the formalsense. Then, according to the paper, we corrected the terms.) and theregularization method are utilized. That is, methods (I-1), (I-2), and(I-3) can be equipped.

On the method (I-3), for instance, the next equation holds for unknown3D displacement vector (ux,uy,uz)^(T) at each point (X,Y,Z) at time t=T:

$\left. {{\theta_{cc}\left( {0\text{,}0\text{,}0} \right)} + {\frac{\partial}{\partial x}{\theta_{cc}\left( {x,y,z} \right)}}} \middle| {}_{{x = 0},{y = 0},{z = 0}}{{ux} + {\frac{\partial}{\partial y}{\theta_{cc}\left( {x,y,z} \right)}}} \middle| {}_{{x = 0},{y = 0},{z = 0}}{{uy} + {\frac{\partial}{\partial z}{\theta_{cc}\left( {x,y,z} \right)}}} \middle| {}_{{x = 0},{y = 0},{z = 0}}{uz} \right. = 0.$θ_(cc)(X,Y,Z;x,y,z) is the 3D phase distribution (x,y,z) of the complexcross-correlation function Cc(X,Y,Z;x,y,z) of the point (X,Y,Z)evaluated from rf echo signals with respect to transmitted ultrasoundpulses at the time t=T and t=T+ΔT:θ_(cc)(X,Y,Z;x,y,z)=tan⁻¹(Im[Cc(X,Y,Z;x,y,z)]/Re[Cc(X,Y,Z;x,y,z)]),where the coordinate (x,y,z) has the origin at (X,Y,Z). In the SOI,occasionally also in time direction, these equations hold (simultaneousequations), and can be solved by least squared method, where, freely,regularization method can be applied (the temporal and spatial magnitudeof the unknown displacement vector distribution, temporal and spatialcontinuity and differentiability of the unknown displacement vectordistribution). Thus, the displacement vector distribution (series) canbe obtained. The gradients of the phase θ_(cc)(X,Y,Z;x,y,z) can beobtained by finite difference approximating or differential filtering.However, for instance, x partial derivative∂/∂x·θ_(cc)(x,y,z)|_(x=0,y=0,z=0) can be obtained as:{Re[Cc(X,Y,Z;0,0,0)]×∂/∂x·Im[Cc(X,Y,Z;x,y,z)]|_(x=0, y=0, z=0)−∂/∂x·Re[Cc(X,Y,Z;x,y,z)]|_(x=0, y=0, z=0)×Im[Cc(X,Y,Z;0,0,0)]}/{Re[Cc(X,Y,Z;0,0,0)]² +Im[Cc(X,Y,Z;0,0,0)]²}.∂/∂x·Re[Cc(X,Y,Z;x,y,z)]|_(x=0, y=0, z=0) can be obtained by finitedifference approximating or differential filtering. Freely, the phases,the signal components, or numerator and denominator can be movingaveraged or low-pass filtered in the time domain.

For instance, the next equation holds for unknown 2D displacement vector(ux,uy)^(T) at each point (X,Y,Z) at time t=T:

$\left. {{\theta_{cc}\left( {0,0} \right)} + {\frac{\partial}{\partial x}{\theta_{cc}\left( {x,y} \right)}}} \middle| {}_{{x = 0},{y = 0}}{{ux} + {\frac{\partial}{\partial y}{\theta_{cc}\left( {x,y} \right)}}} \middle| {}_{{x = 0},{y = 0}}{uy} \right. = 0.$θ_(cc)(X,Y,Z;x,y) is the 2D phase distribution (x,y) of the complexcross-correlation function Cc(X,Y,Z;x,y) of the point (X,Y,Z) evaluatedfrom rf echo signals with respect to transmitted ultrasound pulses atthe time t=T and t=T+ΔT. Method (I-3) can also be applied to measurementof one displacement component distribution.

On the method (I-1), utilized is phase of complex cross-correlationfunction signal in beam direction or scan direction. The next equationholds for unknown displacement component ux at each point (X,Y,Z) attime t=T (the auto-correlation function method's equation):

$\left. {{\theta_{cc}(0)} + {\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{cc}(x)}}} \middle| {}_{x = 0}{ux} \right. = 0.$θ_(cc)(X,Y,Z;x) is the 1D phase distribution (x) of the complexcross-correlation function Cc(X,Y,Z;x) of the point (X,Y,Z) evaluatedfrom rf echo signals with respect to transmitted ultrasound pulses atthe time t=T and t=T+ΔT.

In the ROI, by solving this equation for unknown displacement componentux at each point, the displacement component distribution (series) canbe obtained.

On the method (I-2), in the ROI, occasionally also in time direction,this equation holds in beam direction or scan direction, and the derivedsimultaneous equations can be solved by least squared method, where,freely, regularization method can be applied (the temporal and spatialmagnitude of the unknown displacement component distribution, temporaland spatial continuity and differentiability of the unknown displacementcomponent distribution). Thus, the displacement component distribution(series) can be obtained.

On the methods (I-3) and (I-2), occasionally the unknown displacementvector and the unknown displacement component are dealt as locallyuniform ones. That is, occasionally, under the assumption that the localregion uniformly moves, the simultaneous equations hold for the unknownlocal displacement vector or the unknown local displacement component.Otherwise, occasionally, the simultaneous equations hold under theassumption that the displacement is uniform for temporal finiteinterval. Thus, the spatial distribution (series) can be obtained.

The next method can be equipped. That is, the strain tensor componentcan be directly obtained from spatial derivative of the no time delayphase θ_(cc)(x,y,z;0,0,0)=tan⁻¹(Im[Cc(x,y,z;0,0,0)]/Re[Cc(x,y,z;0,0,0)])of the 3D complex cross-correlation function, of the no time delay phaseθ_(cc)(x,y,z;0,0)=tan⁻¹(Im[Cc(x,y,z;0,0)]/Re[Cc(x,y,z;0,0)]) of the 2Dcomplex cross-correlation function (including beam direction or not), orof the no time delay phaseθ_(cc)(x,y,z;0)=tan⁻¹(Im[Cc(x,y,z;0)]/Re[Cc(x,y,z;0)]) of the 1D complexcross-correlation function (beam direction or scan direction) of thepoint (x,y,z) evaluated from rf echo signals with respect to transmittedultrasound pulses at the time t=T and t=T+ΔT.

For instance, the normal strain component εxx in x axis direction (R=x)at time t=T and at position (X,Y,Z) can be obtained as:

$\begin{matrix}{\begin{matrix}{ɛ\;{xx}} \\\left( {X,Y,Z,T} \right)\end{matrix} = \left. {\frac{\partial}{\partial x}{u_{x}\left( {x,y,z,t} \right)}} \right|_{{x = X},{y = Y},{z = Z},{t = T}}} \\{= \left. {\frac{1}{s_{R}\pi}{\frac{\mathbb{d}}{\mathbb{d}x}\left\lbrack {\frac{c_{x}}{f_{0_{x}}}{\theta_{cc}\left( {x,y,z,t} \right)}} \right\rbrack}} \right|_{{x = X},{y = Y},{z = Z},{t = T}}} \\{= \left. {\frac{1}{s_{R}\pi}\left( {\frac{c_{x}}{f_{0_{x}}}\frac{\begin{matrix}\begin{matrix}\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Cc}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \cdot} \\{{\frac{\mathbb{d}}{\mathbb{d}x}\left\lbrack {{Im}\mspace{11mu}\left\lbrack {{Cc}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \right\rbrack} -}\end{matrix} \\{\frac{\mathbb{d}}{\mathbb{d}x}\left\lbrack {{{Re}\mspace{11mu}\left\lbrack {{Cc}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack} \cdot} \right.}\end{matrix} \\{{Im}\mspace{11mu}\left\lbrack {{Cc}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}\end{matrix}}{\begin{matrix}{{{Re}\mspace{11mu}\left\lbrack {{Cc}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2} +} \\{{Im}\mspace{11mu}\left\lbrack {{Cc}\mspace{11mu}\left( {x,y,z,t} \right)} \right\rbrack}^{2}\end{matrix}}} \right)} \middle| {}_{{x = X},{y = Y},{z = Z},{t = T}}. \right.}\end{matrix}$c_(R) is the ultrasound propagating velocity and scan velocityrespectively when R axis is beam axis and scan axis. f_(0R) isultrasound carrier frequency and sine frequency respectively when R axisis beam axis and scan axis. s_(R) is 4.0 and 2.0 respectively when Raxis is beam axis and scan axis. As above-described, spatial gradient ofthe phase θ_(cc)(x,y,z,t) can also be obtained by finite differenceapproximating or differential filtering after obtaining the phaseθ_(cc)(x,y,z,t) Freely, the phases, the signal components, or numeratorand denominator can be moving averaged or low-pass filtered in the spacedomain. Thus, the strain component distributions (series) can beobtained in the ROI.

The spatial and temporal simultaneous equations of the above-describedequation can be handled, where above-described regularization method canbe applied.

By integrating the partial derivatives of displacement vector componentdistributions (series), the displacement vector distribution (series)can be obtained.

From temporal spatial derivatives of these strain tensor componentdistributions (series) or displacement vector component distributions(series), obtained are strain rate tensor component distributions(series), and acceleration vector component distributions (series).

Freely, (II-1) complex analytic signal method (beam direction or scandirection) is utilized, or (II-2) both of complex analytic signal method(beam direction or scan direction) and the regularization method areutilized, or (II-3) at least 2D distribution (including beam directionor not) of the 3D, 2D phases, or 1D phase of respective of 3D complexsignals with single-octant spectra, 2D complex signals withsingle-quadrant spectra, and conventional 1D complex analytic signal andthe regularization method are utilized (Optical flow algorithm isapplied to the phase of the complex signal.). That is, methods (II-1),(II-2), and (II-3) can be equipped.

On the method (II-3), for instance, the next equation holds for unknown3D displacement vector (ux,uy,uz)^(T) at each point (X,Y,Z) at time t=T:

$\left. {\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{A}\left( {x,y,z,t} \right)}} \middle| {}_{{x = X},{y = Y},{z = Z},{t = T}}{{ux} + {\frac{\mathbb{d}}{\mathbb{d}y}{\theta_{A}\left( {x,y,z,t} \right)}}} \middle| {}_{{x = X},{y = Y},{z = Z},{t = T}}{{uy} + {\frac{\mathbb{d}}{\mathbb{d}z}{\theta_{A}\left( {x,y,z,t} \right)}}} \middle| {}_{{x = X},{y = Y},{z = Z},{t = T}}{{uz} + {\frac{\mathbb{d}}{\mathbb{d}t}{\theta_{A}\left( {x,y,z,t} \right)}}} \middle| {}_{{x = X},{y = Y},{z = Z},{t = T}}{\Delta\; t} \right. = 0$(or for unknown 3D velocity vector (vx,vy,vz)^(T):

$\left. {{{\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{A}\left( {x,y,z,t} \right)}}❘_{{x = X},{y = Y},{z = Z},{t = T}}{{{vx} + {\frac{\mathbb{d}}{\mathbb{d}y}{\theta_{A}\left( {x,y,z,t} \right)}}}❘_{{x = X},{y = Y},{z = Z},{t = T}}{{{vy} + {\frac{\mathbb{d}}{\mathbb{d}y}{\theta_{A}\left( {x,y,z,t} \right)}}}❘_{{x = X},{y = Y},{z = Z},{t = T}}{{{vz} + {\frac{\mathbb{d}}{\mathbb{d}t}{\theta_{A}\left( {x,y,z,t} \right)}}}❘_{{x = X},{y = Y},{z = Z},{t = T}}}}}} = 0} \right)$θ_(A)(x,y,z,t) is the 3D phase distribution (x,y,z) of the complexsignal A(x,y,z,t)(=Re[A(x,y,z,t)]+jIm[A(x,y,z,t)] of the point (x,y,z)at the time t (Δt: transmitted pulse interval):θ_(A)(x,y,z,t)=tan⁻¹(Im[A(x,y,z,t)]/Re[A(x,y,z,t)]).In the SOI, occasionally also in time direction, these equations hold(simultaneous equations), and can be solved by least squared method,where, freely, regularization method can be applied [the temporal andspatial magnitude of the unknown displacement (velocity) vectordistribution, temporal and spatial continuity and differentiability ofthe unknown displacement (velocity) vector distribution]. Thus, thedisplacement (velocity) vector distribution (series) can be obtained.The temporal and spatial gradients of the phase θ_(A)(x,y,z,t) can beobtained by finite difference approximating or differential filtering.However, for instance, x partial derivative∂/∂x·θ_(A)(x,y,z,t)|_(x=X,y=Y,z=Z,t=T) can be obtained as{Re[A(X,Y,Z,T)]×∂/∂x·Im[A(x,y,z,t)]|_(x=X, y=Y, z=Z, t=T)−∂/∂x·Re[A(x,y,z,t)]|_(x=x, y=Y, z=Z, t=T)×Im[A(X,Y,Z,T)]}/{Re[A(X,Y,Z;T)]²+Im[A(X,Y,Z,T)]²}.

∂/∂x·Re[A(x,y,z,t)]|_(x=X, y=Y, z=Z, t=T) can be obtained by finitedifference approximating or differential filtering. Freely, the phases,the signal components, or numerator and denominator can be movingaveraged or low-pass filtered in the time domain.

For instance, the next equation holds for unknown 2D displacement vector(ux,uy)^(T) at each point (X,Y,Z) at time t=T:

${{\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{A}\left( {x,y,t} \right)}}❘_{{x = X},{y = Y},{t = T}}{{{ux} + {\frac{\mathbb{d}}{\mathbb{d}y}{\theta_{A}\left( {x,y,t} \right)}}}❘_{{x = X},{y = Y},{t = T}}{{{uy} + {\frac{\mathbb{d}}{\mathbb{d}t}{\theta_{A}\left( {x,y,t} \right)}}}❘_{{x = X},{y = Y},{t = T}}{\Delta\; t}}}} = 0$(or for unknown 2D velocity vector (vx,vy)^(T):

$\left. {{{\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{A}\left( {x,y,t} \right)}}❘_{{x = X},{y = Y},{t = T}}{{{vx} + {\frac{\mathbb{d}}{\mathbb{d}y}{\theta_{A}\left( {x,y,t} \right)}}}❘_{{x = X},{y = Y},{t = T}}{{{vy} + {\frac{\mathbb{d}}{\mathbb{d}t}{\theta_{A}\left( {x,y,t} \right)}}}❘_{{x = X},{y = Y},{t = T}}}}} = 0} \right).$

θ_(A) (x,y,t) is the 2D phase distribution (x,y) of the complex signalA(x,y,t) (=Re[A(x,y,t)]+jIm[A(x,y,t)] of the point (x,y) at the time t(Δt: transmitted pulse interval):θ_(A)(x,y,z,t)=tan⁻¹ (Re[A(x,y,z,t)]/Im[A(x,y,z,t)]).

Method (II-3) can also be applied to measurement of one displacementcomponent distribution.

On the method (II-1), utilized is phase of complex signal in beamdirection or scan direction. The next equation holds for unknowndisplacement component ux at each point (X,Y,Z) at time t=T:

${\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{A}\left( {x,t} \right)}{_{{x = X},{t = T}}{{ux} + {\frac{\mathbb{d}}{\mathbb{d}t}{\theta_{A}\left( {x,t} \right)}}}}_{{x = X},{t = T}}\Delta\; t} = 0$(or for unknown velocity component vx (the Doppler's equation):

$\left. {{\frac{\mathbb{d}}{\mathbb{d}x}{\theta_{A}\left( {x,t} \right)}{_{{x = X},{t = T}}{{vx} + {\frac{\mathbb{d}}{\mathbb{d}t}{\theta_{A}\left( {x,t} \right)}}}}_{{x = X},{t = T}}\Delta\; t} = 0} \right).$

θ_(A)(x,t) is the 1D phase distribution (x) of the complex signal A(x,t)of the point (x) at the time t (ΔT: transmitted pulse interval).

In the ROI, by solving this equation for unknown displacement componentux (unknown velocity component vx) at each point, the displacement(velocity) component distribution (series) can be obtained.

On the method (II-2), in the ROI, occasionally also in time direction,this equation holds in beam direction or scan direction, and the derivedsimultaneous equations can be solved by least squared method, where,freely, regularization method can be applied (the temporal and spatialmagnitude of the unknown displacement component distribution, temporaland spatial continuity and differentiability of the unknown displacementcomponent distribution). Thus, the displacement component distribution(series) can be obtained.

On the methods (II-3) and (II-2), occasionally the unknown displacement(velocity) vector and the unknown displacement (velocity) vectorcomponent are dealt as locally uniform ones. That is, occasionally,under the assumption that the local region uniformly moves, thesimultaneous equations hold for the unknown local displacement(velocity) vector or the unknown local displacement (velocity)component. Otherwise, occasionally, the simultaneous equations holdunder the assumption that the displacement (velocity) is uniform fortemporal finite interval. Thus, the spatial distribution (series) can beobtained.

The displacement vector distribution (series) can also be obtained byintegrating the obtained velocity vector component distributions(series), or by multiplying transmitted pulse interval Ts to theobtained velocity vector component distributions (series).

From temporal spatial derivatives of these velocity vector distribution(series) or displacement vector distribution (series), obtained arestrain tensor component distributions (series), strain tensor ratecomponent distributions (series), and acceleration vector componentdistributions (series).

There are other various methods for estimating remaining estimationerror vector. These methods can also be utilized in the same way. Whenestimation error of the displacement vector or remaining estimationerror vector is detected during the iterative estimation a priori as thepoint of time-space magnitude and time-space continuity, for instance,the estimate can be cut by compulsion such that the estimate ranges fromthe given smallest value to the given largest value, or such that thedifference between the estimates of the neighboring points settle withinthe given ranges.

As explained, on this conduct form, by iterative estimation themeasurement accuracy can be improved of the displacement vector in the3D SOI, particularly, 3D displacement vector, obtained from thecross-spectrum phase gradient etc. of the ultrasound echo signalsacquired as the responses to more than one time transmitted ultrasound.The local echo signal can be shifted by multiplying complex exponential,or interporation can be performed after shifting sampling ultrasoundsignal. The present invention can improve measurement accuracy oflateral displacements (orthogonal directions to beam direction).Furthermore, the present invention can simplify calculation process intoone without unwrapping the cross-spectrum phase nor utilizingcross-correlation method in order to reduce calculation amount andshorten calculation time.

Moreover, on this conduct form, large displacement (vector) and largestrain (tensor) can be accurately measured by tracking ultrasound echosignals of targeted tissue using echo signal phase as the index (thelocal echo signal can be shifted by multiplying complex exponential, orinterporation can be performed after shifting sampling ultrasoundsignal.), and by adding successively estimated at least more than twodisplacements.

Furthermore, on this conduct form, elastic constant and visco elasticconstant can be accurately measured with high freedom of configurationsof displacement (strain) sensors, mechanical sources, reference regions(mediums).

Next, explains are about elasticity and visco-elasticity constantsmeasurement apparatus related to one of conduct forms of the presentinvention. The elasticity and visco-elasticity constants measurementapparatus related to this conduct form utilize the apparatus shown inFIG. 1 (same as that of the above-explained displacement vector andstrain measurement), and the apparatus measures elastic constants andvisco elastic constants from displacement vector, strain tensor, etc.measured by using the above-explained displacement and strainmeasurement method.

At first, the assumptions are explained of the elasticity andvisco-elasticity constants measurement apparatus related to this conductform. The following constants are assumed to be measured only in thetarget ROI (SOI) set in the measurement object, elastic constants suchas shear modulus, Poisson's ratio, etc., visoc elastic constants such asvisco shear modulus, visco Poisson's ratio, etc., delay times orrelaxation times relating these elastic constants and visco elasticconstants, or density. All the mechanical sources are assumed to existoutside of the ROI. Then, if there exist other mechanical sources inaddition to set mechanical sources or if the mechanical sources areuncontrollable, the following constants can be measured in the targetROI (SOI), elastic constants such as shear modulus, Poisson's ratio,etc., visoc elastic constants such as visco shear modulus, viscoPoisson's ratio, etc., delay times or relaxation times relating theseelastic constants and visco elastic constants, or density. Neitherinformation is needed about mechanical sources, such as positions, forcedirections, force magnitudes, etc. Moreover neither stress data norstrain data are needed at the target body surface. Only the ROI ismodeled using finite difference method or finite element method.

If the mechanical sources originally exist near the ROI, only themechanical sources can be utilized. In the case of observation of livingtissues, for instance, such mechanical sources include normallyuncontrollable mechanical sources such as heart motion, respiratorymotion, blood vessel motion, body motion. In general, lung, air, bloodvessel, blood are included in the ROI. In this case, without disturbingthe deformation field, the following constants can be measured, i.e.,elastic constants such as shear modulus, Poisson's ratio, etc., visocelastic constants such as visco shear modulus, visco Poisson's ratio,etc., delay times or relaxation times relating these elastic constantsand visco elastic constants, or density. This is effective particularlywhen the ROI deeply situates.

When solving the first order partial differential equations, as initialconditions the following can be utilized, i.e., reference shear modulusand reference Poisson's ratio for elastic constants, reference viscoshear modulus and reference visco Poisson's ratio for visco elasticconstants, reference density for density. In this case, referencemediums or reference regions are set in the original ROI or near theoriginal ROI, after which the final ROI is set such that the final ROIincludes the original ROI as well as the references. By measuring in theROI including reference regions strain tensor field, strain rate tensorfield, and acceleration vector field, the references are realized.

The size and the position of the reference mediums or reference regionsshould be set such that they should widely cross the direction of thedominant tissue deformation. For instance, if the mechanical source haslarge contact surface, large reference region must be set. Otherwise, ifthe mechanical source has small contact surface, by setting thereference region near the mechanical source, small reference region canbe used. The estimates can be also used as their references.

The present invention can provide absolute shear modulus distribution,relative shear modulus distribution, absolute Poisson's ratiodistribution, relative Poisson's ratio distribution, absolute viscoshear modulus distribution, relative visco shear modulus distribution,absolute visco Poisson's ratio distribution, relative visco Poisson'sratio distribution, absolute or relative delay time distributionsrelating these elastic constants and visco elastic constants, orabsolute or relative relaxation time distributions relating theseelastic constants and visco elastic constants, absolute densitydistribution, or relative density distribution. Here, distributions ofreference Poisson's ratio, reference visco Poisson's ratio, referencedensity must be distributions of absolute values, while distributions ofother reference elastic constants, and reference visco elastic contantsmay be distributions of relative values.

As the numerical solution method of the first order partial differentialequations, finite difference method or finite element method can beutilized. By utilizing the regularized algebraic equations, if thestrain tensor field data is contaminated with errors (noises), or if thereference medium or reference region is small, or if the referenceposition is ill-conditioned, the following distribution can be stablyestimated, i.e., shear modulus distribution, Poisson's ratiodistribution, visco shear modulus distribution, visco Poisson's ratio,density, etc.

Referring to FIG. 1 again, next explain is about the means of dataprocessing 1, i.e., calculation method of shear modulus distribution,Poisson's ratio distribution, visco shear modulus distribution, viscoPoisson's ratio distribution, delay time distributions, relaxation timedistributions, or density distribution, etc. When the 3D strain tensor,the 3D strain rate tensor, the 3D acceleration vector, etc. aremeasured, for instance, on the Cartesian coordinate system (x,y,z), thenext simultaneous first order partial equations from (125) to (137″) aredealt with, where the shear modulus distribution μ, the Poisson's ratiodistribution ν, the visco shear modulus distribution μ′, the viscoPoisson's ratio distribution ν′, the delay time distributions τ, therelaxation time distributions τ′, the strain tensor field ε, the strainrate tensor field ε′.

That is, when the 3D strain tensor is measured, and only the shearmodulus distribution μ is unknown, the next equations are dealt with,

$\begin{matrix}{{{{\left\{ {{\phi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}\left( {\ln\;\mu} \right)_{,j}} + \left\{ {{\phi\; ɛ_{aa}\delta_{ij}} + ɛ_{ij}} \right\}_{,j}} = 0},} & (125) \\{{{{where}\mspace{14mu}\phi} = \frac{v}{1 - {2v}}},{or}} & \left( 125^{\prime} \right) \\{{{{\left\{ {{\phi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}\mu_{,j}} + {\left\{ {{\phi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}_{,j}\mu}} = 0},} & (126) \\{{{where}\mspace{14mu}\phi} = {\frac{v}{1 - {2v}}.}} & \left( 126^{\prime} \right)\end{matrix}$

When the 3D strain tensor is measured, and the shear modulusdistribution μ and the Poisson's ratio distribution ν are unknown, thenext equations are dealt with,

$\begin{matrix}{{{{\left\{ {ɛ_{\alpha\alpha}\delta_{ij}} \right\}\lambda_{,j}} + {\left\{ {ɛ_{\alpha\alpha}\delta_{ij}} \right\}_{,j}\lambda} + {2ɛ_{ij}\mu_{,j}} + {2ɛ_{{ij},j}\;\mu}} = 0},} & (127) \\{{{where}\mspace{14mu}\lambda} = {\frac{2v}{1 - {2v}}{\mu.}}} & \left( 127^{\prime} \right)\end{matrix}$

When the 3D strain tensor and the 3D strain rate tensor are measured,and the shear modulus distribution μ and the visco shear modulusdistribution μ′ are unknown, the next equations are dealt with,

$\begin{matrix}{{{{\left\{ {{\phi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\} u_{,j}} + {\left\{ {{\phi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}_{,j}\mu} + {\left\{ {{\phi^{\prime}ɛ_{\alpha\alpha}^{\prime}\delta_{ij}} + ɛ_{ij}^{\prime}} \right\}\mu_{,j}^{\prime}} + {\left\{ {{\phi^{\prime}ɛ_{\alpha\alpha}^{\prime}\delta_{ij}} + ɛ_{ij}^{\prime}} \right\}_{,j}\mu^{\prime}}} = 0},} & (128) \\{{{{where}\mspace{20mu}\phi} = \frac{v}{1 - {2v}}},} & \left( 128^{\prime} \right) \\{{\phi^{\prime} = \frac{v^{\prime}}{1 - {2v^{\prime}}}},\mspace{14mu}{or}} & \left( 128^{\prime\prime} \right) \\{{{\left\lbrack {\int_{t^{\prime}}^{t}{{\phi\left( {t - \tau} \right)}{\mu\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\phi\left( {t - \tau} \right)\mu\left( {t - \tau} \right)}{{\phi^{\prime}\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}}\left( {t - \tau} \right)} \right\}{ɛ_{\alpha\alpha}^{\prime}(\tau)}\ {\mathbb{d}\tau}\;\delta_{ij}}} \right\rbrack_{,j} + \left\lbrack {\int_{t^{\prime}}^{t}{{\mu\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\mu\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{ij}^{\prime}(\tau)}\ {\mathbb{d}\tau}}} \right\rbrack_{,j}} = 0},} & \left( 128^{\prime\prime\prime} \right)\end{matrix}$where t′ is initial time. If either the shear modulus distribution μ orthe visco shear modulus distribution μ′ is given, the next equations canbe dealt with,{φε_(αα)δ_(ij)+ε_(ij)}μ={φ′ε′_(αα)δ_(ij)+ε′_(ij)}μ′.  (128″″)

If both the shear modulus distribution μ and the visco shear modulusdistribution μ′ are unknown, from this equations, the relaxation timeμ′(t)/μ(t) can be calculated, and can be utilized in the above equations(128′″).

When the 3D strain tensor and the 3D strain rate tensor are measured,and the shear modulus distribution μ, the Poisson's ratio distributionν, the visco shear modulus distribution μ′, and the visco Poisson'sratio distribution ν′ are unknown, the next equations are dealt with,

$\begin{matrix}{{{{\left\{ {ɛ_{\alpha\alpha}\delta_{ij}} \right\}\lambda_{,j}} + {\left\{ {ɛ_{\alpha\alpha}\delta_{ij}} \right\}_{,j}\lambda} + {2\; ɛ_{ij}\mu_{,j}} + {2ɛ_{{ij},j}\mu} + {\left\{ {ɛ_{\alpha\alpha}^{\prime}\delta_{ij}} \right\}\lambda_{,j}^{\prime}} + {\left\{ {ɛ_{\alpha\alpha}^{\prime}\delta_{ij}} \right\}_{,j}\lambda^{\prime}} + {2\; ɛ_{ij}^{\prime}\mu_{,j}^{\prime}} + {2\; ɛ_{{ij},j}^{\prime}\mu^{\prime}}} = 0},} & (129) \\{{{{where}\mspace{20mu}\lambda} = {\frac{2v}{1 - {2v}}\mu}},} & \left( 129^{\prime} \right) \\{{\lambda^{\prime} = {\frac{2v^{\prime}}{1 - {2v^{\prime}}}\mu^{\prime}}},{or}} & \left( 129^{\prime\prime} \right) \\{{{\left\lbrack {\int_{t^{\prime}}^{t}{{\lambda\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\lambda\left( {t - \tau} \right)}{\lambda^{\prime}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{\alpha\alpha}^{\prime}(\tau)}\ {\mathbb{d}\tau}\;\delta_{ij}}} \right\rbrack_{,j} + {2\left\lbrack {\int_{t^{\prime}}^{t}{{\mu\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\mu\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{ij}^{\prime}(\tau)}\ {\mathbb{d}\tau}}} \right\rbrack}_{,j}} = 0},} & \left( 129^{\prime\prime\prime} \right)\end{matrix}$where t′ is initial time. Either both the shear modulus distribution μand the visco shear modulus distribution μ′ or both the Poisson's ratiodistribution ν and visco Poisson's ratio distribution ν′ are given, thenext equations can be dealt with,λε_(αα)δ_(ij)+2ε_(ij)μ=λ′ε′_(αα)δ_(ij)+2ε′_(ij)μ′.  (129″″)

From this equations, the relaxation time μ′(t)/μ(t) can always becalculated. Then if either the shear modulus distribution μ or the viscoshear modulus distribution μ′ is given, the obtained shear modulusdistribution μ and the obtained visco shear modulus distribution μ′ canbe utilized in the above equations (129′″). Otherwise, if either thePoisson's ratio distribution ν or the visco Poisson's ratio distributionν′ is given, the obtained Poisson's ratio distribution ν, the obtainedvisco Poisson's ratio distribution ν′, and the obtained relaxation timeλ′(t)/λ(t) can be utilized in the above equations (129′″).

Equations (128′″), (128″″), (129′″), and (129″″) can be dealt with whenthe target is fluid such as water, secretions, blood, etc., or tissueincludes the fluid much. The equations can also be dealt with afterfirst temporal partial differentiation or partial integration.Theoretically, the elastic constant distributions and visco elasticconstant distributions need to be invariant from the initial time t′ totime t.

When the 2D strain tensor, the 2D strain rate tensor, etc. are measured,the simultaneous first order partial equations from (125) to (129″″) [i,j=1,2] or the next simultaneous first order partial equations from (130)to (134″″) [i, j=1,2] are dealt with. The equations from (125) to(129″″) [i, j=1,2] are dealt with approximately under plane straincondition, while the equations from (130) to (134″″) [i, j=1,2] aredealt with approximately under plane stress condition.

When the 2D strain tensor is measured, and only the shear modulusdistribution μ is unknown, the next equations are dealt with,

$\begin{matrix}{{{{\left\{ {{\varphi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}\left( {\ln\;\mu} \right)_{,j}} + \left\{ {{\varphi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}_{,j}} = 0},} & (130) \\{{{{where}\mspace{14mu}\varphi} = \frac{v}{1 - v}},{or}} & \left( 130^{\prime} \right) \\{{{{\left\{ {{\varphi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}\mu_{,j}} + {\left\{ {{\varphi\; ɛ_{\alpha\alpha}\delta_{ij}} + ɛ_{ij}} \right\}_{,j}\mu}} = 0},} & (131) \\{{{where}\mspace{14mu}\varphi} = {\frac{v}{1 - v}.}} & \left( 131^{'} \right)\end{matrix}$When the 2D strain tensor is measured, and the shear modulusdistribution μ and the Poisson's ratio distributions are unknown, thenext equations are dealt with,

$\begin{matrix}{{{{\left\{ {ɛ_{\alpha\alpha}\delta_{i\; j}} \right\}\gamma_{,j}} + {\left\{ {ɛ_{\alpha\alpha}\delta_{i\; j}} \right\}_{,j}\gamma} + {ɛ_{i\; j}\mu_{,j}} + {ɛ_{{i\; j},j}\mu}} = 0},} & (132) \\{{{where}\mspace{14mu}\gamma} = {\frac{v}{1 - v}{\mu.}}} & \left( 132^{\prime} \right)\end{matrix}$

When the 2D strain tensor and the 2D strain rate tensor are measured,and the shear modulus distribution μ and the visco shear modulusdistribution μ′ are unknown, the next equations are dealt with,

$\begin{matrix}{{{{\left\{ {{{\varphi ɛ}_{\alpha\alpha}\delta_{i\; j}} + ɛ_{i\; j}} \right\}\mu_{,j}} + {\left\{ {{{\varphi ɛ}_{\alpha\alpha}\delta_{i\; j}} + ɛ_{i\; j}} \right\}_{,j}\mu} + {\left\{ {{\varphi^{\prime}ɛ_{\alpha\alpha}^{\prime}\delta_{i\; j}} + ɛ_{i\; j}^{\prime}} \right\}\mu_{,j}^{\prime}} + {\left\{ {{\varphi^{\prime}ɛ_{\alpha\alpha}^{\prime}\delta_{i\; j}} + ɛ_{i\; j}^{\prime}} \right\}_{,j}\mu^{\prime}}} = 0},} & (133) \\{{{{where}\mspace{14mu}\varphi} = \frac{v}{1 - v}},} & \left( 133^{\prime} \right) \\{{\varphi^{\prime} = \frac{v^{\prime}}{1 - v^{\prime}}},{or}} & \left( 133^{\prime\prime} \right) \\{{{\begin{bmatrix}{\int_{t^{\prime\;}}^{t}{{\varphi\left( {t - \tau} \right)}{\mu\left( {t - \tau} \right)}\exp}} \\{\left\{ {{- \frac{\varphi\left( {t - \tau} \right){\mu\left( {t - \tau} \right)}}{{\varphi^{\prime\;}\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}}\left( {t - \tau} \right)} \right\}{ɛ_{\alpha\alpha}^{\prime}(\tau)}{\mathbb{d}{\tau\delta}_{i\; j}}}\end{bmatrix}_{,j} + \left\lbrack {\int_{t^{\prime\;}}^{t}{{\mu\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\mu\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{i\; j}^{\prime}(\tau)}{\mathbb{d}\tau}}} \right\rbrack_{,j}} = 0},} & \left( 133^{\prime\prime\prime} \right)\end{matrix}$where t′ is initial time. If either the shear modulus distribution μ orthe visco shear modulus distribution μ′ is given, the next equations canbe dealt with,{φε_(αα)δ_(ij)+ε_(ij)}μ={φ′ε′_(αα)δ_(ij)+ε′_(ij)}μ′.  (133″″)

If both the shear modulus distribution μ and the visco shear modulusdistribution μ′ are unknown, from this equations, the relaxation timeμ′(t)/μ(t) can be calculated, and can be utilized in the above equations(133′″).

When the 2D strain tensor and the 2D strain rate tensor are measured,and the shear modulus distribution μ, the Poisson's ratio distributionν, the visco shear modulus distribution μ′, and the visco Poisson'sratio distribution ν′ are unknown, the next equations are dealt with,

$\begin{matrix}{{{{\left\{ {ɛ_{\alpha\alpha}\delta_{i\; j}} \right\}\gamma_{,j}} + {\left\{ {ɛ_{\alpha\alpha}\delta_{{i\; j}\;}} \right\}_{,j}\gamma} + {ɛ_{i\; j}\mu_{,j}} + {ɛ_{{i\; j},j}\mu} + {\left\{ {ɛ_{\alpha\alpha}^{\prime}\delta_{i\; j}} \right\}\gamma_{,j}^{\prime}} + {\left\{ {ɛ_{\alpha\alpha}^{\prime}\delta_{i\; j}} \right\}_{,j}\gamma^{\prime}} + {ɛ_{i\; j}^{\prime}\mu_{,j}^{\prime}} + {ɛ_{{i\; j},j}^{\prime}\mu^{\prime}}} = 0},} & (134) \\{{{{where}\mspace{14mu}\gamma} = {\frac{v}{1 - v}\mu}},} & \left( 134^{\prime} \right) \\{{\gamma^{\prime} = {\frac{v^{\prime}}{1 - v^{\prime}}\mu^{\prime}}},{or}} & \left( 134^{\prime\prime} \right) \\{{{\begin{bmatrix}{\int_{t^{\prime\;}}^{t}{{\gamma\left( {t - \tau} \right)}\exp}} \\{\left\{ {{- \frac{\gamma\left( {t - \tau} \right)}{\gamma^{\prime\;}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{\alpha\alpha}^{\prime}(\tau)}{\mathbb{d}{\tau\delta}_{i\; j}}}\end{bmatrix}_{,j} + \left\lbrack {\int_{t^{\prime\;}}^{t}{{\mu\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\mu\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{i\; j}^{\prime}(\tau)}{\mathbb{d}\tau}}} \right\rbrack_{,j}} = 0},} & \left( 134^{\prime\prime\prime} \right)\end{matrix}$where t′ is initial time. Either both the shear modulus distribution μand the visco shear modulus distribution μ′ or both the Poisson's ratiodistribution ν and visco Poisson's ratio distribution ν′ are given, thenext equations can be dealt with,γε_(αα)δ_(ij)+ε_(ij)μ=γ′ε′_(αα)δ_(ij)+ε′_(ij)μ′.  (134″″)

From this equations, the relaxation time μ′(t)/μ(t) can always becalculated. Then if either the shear modulus distribution μ or the viscoshear modulus distribution μ′ is given, the obtained shear modulusdistribution μ and the obtained visco shear modulus distribution μ′ canbe utilized in the above equations (134′″). Otherwise, if either thePoisson's ratio distribution ν or the visco Poisson's ratio distributionν′ is given, the obtained Poisson's ratio distribution ν, the obtainedvisco Poisson's ratio distribution ν′, and the obtained relaxation timeγ′(t)/γ(t) can be utilized in the above equations (134′″).

Equations (133′″), (133″″), (134′″), and (134″″) can be dealt with whenthe target is fluid such as water, secretions, blood, etc., or tissueincludes the fluid much. The equations can also be dealt with afterfirst temporal partial differentiation or partial integration.Theoretically, the elastic constant distributions and visco elasticconstant distributions need to be invariant from the initial time t′ totime t.

When the 1D strain tensor, the 1D strain rate tensor, etc. are measured,the simultaneous first order partial equations from (135) to (137″) aredealt with.

When the 1D strain tensor is measured, and only the shear modulusdistribution μ is unknown, the next equations are dealt with,ε₁₁(ln μ)_(,1)+ε_(11,1)=0,  (135)orε₁₁μ_(,1)+ε_(11,1)μ=0.  (136)

When the 1D strain tensor and the 1D strain rate tensor are measured,and the shear modulus distribution μ and the visco shear modulusdistribution μ′ are unknown, the next equations are dealt with,

$\begin{matrix}{{{{ɛ_{11}\mu_{,1}} + {ɛ_{11,1}\mu} + {ɛ_{11}^{\prime}\mu_{,1}^{\prime}} + {ɛ_{11,1}^{\prime}\mu^{\prime}}} = 0},{or}} & (137) \\{{\left\lbrack {\int_{t^{\prime}}^{t}{{\mu\left( {t - \tau} \right)}\exp\left\{ {{- \frac{\mu\left( {t - \tau} \right)}{\mu^{\prime}\left( {t - \tau} \right)}}\left( {t - \tau} \right)} \right\}{ɛ_{11}^{\prime}(\tau)}{\mathbb{d}\tau}}} \right\rbrack_{,1} = 0},} & \left( 137^{\prime} \right)\end{matrix}$where t′ is initial time. If either the shear modulus distribution μ orthe visco shear modulus distribution μ′ is given, the next equations canbe dealt with,ε₁₁μ=ε′₁₁μ′.  (137″)

If both the shear modulus distribution μ and the visco shear modulusdistribution μ′ are unknown, from this equation, the relaxation timeμ′(t)/μ(t) can be calculated, and can be utilized in the above equations(137′).

Equations (137′) and (137″) can be dealt with when the target is fluidsuch as water, secretions, blood, etc., or tissue includes the fluidmuch. The equations can also be dealt with after first temporal partialdifferentiation or partial integration. Theoretically, the shear modulusdistribution and visco shear modulus distribution need to be invariantfrom the initial time t′ to time t.

In the equations (125), (130), (135), changed can be the signs of theterms not including (ln μ),j, and together changed can be (ln μ),j into{ln(1/μ)},j, then resultant partial differential equations can be dealtwith for unknown ln(1/μ). Although regarding with equations (125),(130), (135) unknown ln μ cases are explained below, in unknown ln(1/μ)cases ln μ or μ can be estimated after ln(1/μ) or (1/μ) are estimated insimilar ways.

In the equations (126), (131), (136), changed can be the signs of theterms not including μ,j, and together changed can be μ into (1/μ), thenresultant partial differential equations can be dealt with for unknown(1/μ). Although regarding with equations (126), (131), (136) unknown μcases are explained below, in unknown (1/μ) cases μ or ln μ can beestimated after (1/μ) or ln(1/μ) are estimated in similar ways.

These can be effective when the ROI includes extremely high shearmodulus object such as bone, interstitial needle (for biopsy andtreatment), etc.

When the target is fluid such as water, secretions, blood, etc., ortissue includes the fluid much, in the equations (125), (126), (127),(130), (131), (132), (135), (136) the elastic constants can be changedinto the corresponding visco elastic constants, and the strain tensorcan be changed into the strain rate tensor. Also in this case, in theequations (125), (130), (135), changed can be the signs of the terms notincluding (ln μ′),j, and together changed can be (ln μ′),j into{ln(1/μ′)},j, then resultant partial differential equations can be dealtwith for unknown ln(1/μ′). Although regarding with equations (125),(130), (135) unknown ln μ cases are explained below, in unknown ln(1/μ′)cases ln μ′ or μ′ can be estimated after ln(1/μ′) or (1/μ′) areestimated in similar ways.

In the equations (126), (131), (136), changed can be the signs of theterms not including μ′,j, and together changed can be μ′ into (1/μ′),then resultant partial differential equations can be dealt with forunknown (1/μ′). Although regarding with equations (126), (131), (136)unknown μ′ cases are explained below, in unknown (1/μ′) cases μ′ or lnμ′ can be estimated after (1/μ′) or ln(1/μ′) are estimated in similarways.

These can be effective when the ROI includes extremely high shearmodulus object such as bone, interstitial needle (for biopsy andtreatment), etc.

When elasticity or visco elasticity is anisotropic, correspondinglyderived equations from (125) to (137″) can be dealt with.

Regarding density distribution ρ, measured acceleration vector field ais used. Specifically, in equations (126), (128), (128′″), (131), (132),(133), (133′″), (134), (134′″), (½)ρa_(i) is added to right terms, inequations (127), (129), (129′″) ρa_(i) is added to right terms, and inequations (136), (137), (137′) (⅓)ρa_(i) is added to right term. Theknown density distribution is used in the region, and the unknowndensity distribution is estimated with the unknown shear modulusdistribution μ, the unknown Poisson's ratio distribution ν, the unknownvisco shear modulus distribution μ′, and the unknown visco Poisson'sratio distribution ν′. When the target is fluid such as water,secretions, blood, etc., or tissue includes the fluid much, in theequations (126), (127), (131), (132), (136) the elastic constants can bechanged into the corresponding visco elastic constants, and the straintensor can be changed into the strain rate tensor. The density can notbe handled when partial differential equations (126), (131), (136) aredirectly solved for ln(1/μ), (1/μ), ln(1/μ′), and (1/μ′).

Specifically, according to the measured deformation field, i.e., thestrain tensor field, the strain rate tensor field [when dealing with thedensity ρ (below omitted), the acceleration vector field, the temporalfirst derivative of the acceleration vector field, the strain tensorfield, the strain rate tensor field] and/or the accuracy of the measureddeformation field, dealt with all over the 3D SOI 7 are the simultaneousfirst order partial differential equations from (125) to (129″″), ordealt within the plural 3D SOIs, plural 2D ROIs, plural 1D ROIs set inthe 3D SOI 7 are respectively the simultaneous first order partialdifferential equations from (125) to (129″″), the simultaneous firstorder partial differential equations from (125) to (134″″), the firstorder partial differential equations from (135) to (137″). When pluralindependent deformation fields are measured, according to the accuracyof the measured deformation fields, freely either of the equations from(125) to (137″) or the plural equations of the equations from (125) to(137″) can be dealt with at each point of interest. That is, theseequations are solved individually or simultaneously. The pluralindependent deformation fields can be generated under the differentpositions of the mechanical sources and the reference regions. These 3DSOIs, 2D ROIs, and 1D ROIs can include same regions in the 3D SOI 7.

The Poisson's ratio and visco Poisson's ratio can respectively beapproximated from ratios of the principal values of the strain tensorand strain rate tensor (on 3D measurement, either of three ratios of theprinciple values, or three or two mean values of the ratios). Whenplural deformation fields are measured, the Poisson's ratio and thevisco Poisson's ratio can respectively be approximated as the mean valueof the calculated ones from the plural fields. Typical values can alsobe utilized for the Poisson's ratio and the visco Poisson's ratio. Forinstance, the object is assumed to be incompressible, then the valuesare approximated as the value of about 0.5. Particularly, on equationsfrom (130) to (134″″), the object can be assumed to be completelyincompressible, then the values are approximated as 0.5.

As initial conditions, at least at one reference point, or at least inproperly set one wide reference region, given are reference shearmodulus, reference Poisson's ratio, reference visco shear modulus,reference visco Poisson's ratio, etc.

That is, reference shear moduli (absolute or relative values) are givenat least in one reference region

$\begin{matrix}{{\varpi_{\mu,1}\left( {1 = {1 \sim N_{\mu}}} \right)}.} & \; \\{{{\ln\;{\mu\left( {x,y,z} \right)}} = {\ln\;{\hat{\mu}\left( {x,y,z} \right)}}},\mspace{14mu}{\varpi_{\mu,l} \in \left( {x,y,z} \right)}} & (138) \\{{{\mu\left( {x,y,z} \right)} = {\hat{\mu}\left( {x,y,z} \right)}},\mspace{14mu}{\varpi_{\mu,1} \in \left( {x,y,z} \right)}} & \left( 138^{\prime} \right)\end{matrix}$

That is, reference Poisson's ratios (absolute values) are given at leastin one reference region ω _(ν,l)(l=1˜N_(ν)).ν(x,y,z)={circumflex over (ν)}(x,y,z), ω _(ν,l)ε(x,y,z)  (139)

That is, reference visco shear moduli (absolute or relative values) aregiven at least in one reference region ω _(μ′,l)(l=1˜N_(μ′)).μ′(x,y,z)={circumflex over (μ)}′(x,y,z), ω _(μ′,l)ε(x,y,z)  (140)

That is, reference visco Poisson's ratios (absolute values) are given atleast in one reference region ω _(ν′,l)(l=1˜N_(ν′)).ν′(x,y,z)={circumflex over (ν)}′(x,y,z), ω _(ν′,l)ε(x,y,z)  (141)

When elasticity or visco elasticity is anisotropic, correspondinglyderived equations from (125) to (137″) and correspondingly derivedinitial conditions from (138) to (141) can be dealt with.

On discrete Cartesian's coordinate (x,y,z)˜(IΔx, JΔy, KΔz) in ROI 7finite difference approximation or finite element method based on theGalerkin's method or the variational principle is applied to the shearmodulus distribution μ, the Poisson's ratio distribution ν, the elasticconstant distribution φ, the elastic constant distribution λ, theelastic constant distribution φ, the elastic constant distribution γ,the visco shear modulus distribution μ′, the Poisson's ratiodistribution ν′, the visco elastic constant distribution φ′, the viscoelastic constant distribution λ′, the visco elastic constantdistribution φ′, the visco elastic constant distribution γ′, thedisplacement distribution, the strain distribution, and the strain ratedistribution. Then algebraic equations are derived from the first orderpartial differential equations and initial conditions, and usually thealgebraic equations are normalized, for instance, by the root square ofthe summation of the powers of the spatially inhomogeneous coefficients(or the distributions) multiplied to the shear modulus (distribution) μ,the Poisson's ratio (distribution) ν, the elastic constant(distribution) φ, the elastic constant (distribution) λ, the elasticconstant (distribution) φ, the elastic constant (distribution) γ, thevisco shear modulus (distribution) μ′, the Poisson's ratio(distribution) ν′, the visco elastic constant (distribution) φ′, thevisco elastic constant (distribution) λ′, the visco elastic constant(distribution) φ′, the visco elastic constant (distribution) γ′.Furthermore, the algebraic equations can be regularized. Here, elasticconstants λ and μ are called as Lame's constants, while visco elasticconstants λ′ and μ′ are called as visco Lame's constants.

For instance, finite difference method is utilized, the simultaneousequations are derived.EGs=e  (142)s is unknown vector comprised of the unknown shear modulus distributionμ, the unknown elastic constant distribution λ, the unknown elasticconstant distribution γ, the unknown visco shear modulus distributionμ′, the unknown visco elastic constant distribution λ′, the unknownvisco elastic constant distribution γ′, etc. G is coefficients matrixcomprised of finite approximations of the 3D, 2D or 1D partialderivatives. E and e are respectively matrix and vector comprised ofstrain tensor data, strain rate tensor data, their derivatives, andgiven elastic constants, or visco elastic constants.

Equations (142) is solved by least squares method, where in order toreduce the noises of the measured strain tensor data and strain ratetensor data, the strain distribution and the strain rate distributionare determined as spatially, temporally, or spatio-temporally low passfiltered ones. However, inverse of EG amplifies the high frequencynoises filled with e. Then, s becomes unstable. Thus, to stabilize s theregularization method is applied. Utilizing the regularizationparameters α1 and α2 (at least larger than zero), next equation (143) isminimized with respect to s, where T indicates transpose.error(s)=|e−EGs| ²+α1|Ds| ²+α2|D ^(T) Ds| ²  (143)

D and D^(T)D are respectively 3D, 2D, or 1D gradient and Laplacianoperator of the unknown shear modulus distribution μ, the unknownelastic constant distribution λ, the unknown elastic constantdistribution γ, the unknown visco shear modulus distribution μ′, theunknown visco elastic constant distribution λ′, the unknown viscoelastic constant distribution γ′, etc. That is, with respect to eachunknown distribution, the regularization method can be applied over the3D SOI, plural 3D SOIs, 2D ROIs, or 1D ROIs. As Ds and D^(T)D arepositive definite, error(s) absolutely has one minimum value. Byminimizing error(s), the next regularized normal equations are derived.(G ^(T) E ^(T) EG+α1D ^(T) D+α2D ^(T) DD ^(T) D)s=G ^(T) E ^(T) e  (144)Therefore, the solusion is obtained ass=(G ^(T) E ^(T) EG+α1D ^(T) D+α2D ^(T) DD ^(T) D)⁻¹ G ^(T) E ^(T)e  (145)

When the finite element method is utilized, in similar ways, the leastsquares method and the regularization method are applied to the derivedsimultaneous equations. In this case, G is comprised of basis functionof the unknown nodal elastic modulus distribution and the unknown nodalvisco elastic modulus distribution. Moreover, utilizing theregularization parameter α0 (at least larger than zero), α0|s|² andα0|Gs|² can be added to the equation (143). Furthermore, instead ofα1|Ds|² and α2|D^(T)Ds|, α1|DGs| and α2|D^(T)DGs| can also be utilized.

The regularization parameter of important information is set relativelylarge. Thus, the regularization parameter utilized for each constantdepends on deformation measurement accuracy (SNR), deformation state,configurations of mechanical sources and reference regions, number ofthe utilized independent deformation fields, etc.; then position of theunknown constant, direction of the partial derivative, etc.

From the ratio of each elastic constant E with respect to thecorresponding visco elastic constant E′, i.e., (E′/E), for instance,when measured are the shear modulus, the Poisson's ratio, the Lameconstants, etc. and their corresponding visco elastic modulus, estimatedcan be the time delay distribution τ [case when visco elastic modulus isdetermined from (128), (129), (133), (134), (136), or(137)] orrelaxation time distribution τ′ [case when visco elastic modulus isdetermined from (128′″), (128″″), (129′″), (129″″), (133′″), (133″″),(134′″), (134″″), (136′″), (136″″), (137′), or (137″), or case whenvisco elastic modulus is determined from (125), (126), (127), (130),(131), (132), (135), or (136) where the elastic moduli and strain tensorcomponents are respectively changed into the corresponding visco elasticmouli and the strain rate tensor components]. Moreover, from straintensor data and elastic moduli data, elastic energy distribution can beobtained, while from strain rate tensor data and visco elastic modulidata, consumed energy distribution can be obtained.

These elastic constants and visco elastic constants can be temporallychanged. Thus, the spatial and temporal simultaneous equations of theabove-described equation can be handled, where above-describedregularization method can spatially and temporally be applied.

If the time sequence of the elastic modulus distribution or the viscoelastic modulus distribution is estimated, by spectrum analysis, thedistribution of the frequency variance of the elastic modulus or thevisco elastic modulus can approximately be obtained. Moreover, if thetime sequence of the time delay distribution or the relaxation timedistribution is estimated, by spectrum analysis, the distribution of thefrequency variance of the time delay or the relaxation time canapproximately be obtained. When estimating the distributions of thefrequency variances of these elastic modulus, visco elastic modulus,time delay, relaxation time, the deformation field is measured withchanging the frequency of the mechanical source, or with utilizingbroadband mechanical source. Furthermore, at each time, from straintensor data and elastic moduli data, elastic energy distribution can beobtained, while from strain rate tensor data and visco elastic modulidata, consumed energy distribution can be obtained.

When solving by the iterative method such as the conjugate gradientmethod equations from (143) to (145) derived from equations from (125)to (137″) for each unknown elastic modulus distribution and each unknownvisco elastic modulus distribution, as explained below, if necessary,newly the reference regions are set in the ROI in addition to thepre-described reference regions, and properly initial values of theestimates are set in the unknown region. In general, each initial valueis set based on the a priori information such as homogeneity andinhomogeneity. Thus, calculation amount can be reduced.

Regarding with elasticity distribution, for instance, on 1D measurementbased on the partial differential equation (135) or (136), byanalytically solving these equations, the relative shear modulus of thepoint x=X with respect to the point x=A can be estimated from the ratioof the strains ε(A)/ε(X) (Japanese Patent Application PublicationJP-7-55775). This is effective when tissues deforms in x direction.(Moreover, regarding with visco elasticity distribution, for instance,on 1D measurement based on the partial differential equation (135) or(136), by analytically solving these equations, the relative visco shearmodulus of the point x=X with respect to the point x=A can be estimatedfrom the ratio of the strain rates ε′(A)/ε′(X). Below, the shear modulusis dealt with, for instance.)

However, for instance, in the singular points or the singular regionswhere the strain is numerically zero, or the sign of the strain changes,the shear modulus can be stably estimated with the above-describedregularization method using the absolute reference values or therelative reference values (reference values obtained from ratio of thestrains in addition to given reference values.). Otherwise, in theunknown points or the unknown regions where the absolute strain is lessthan the positive value A (threshold), in a similar way, the shearmodulus can be stably estimated using the absolute reference values orthe relative reference values (reference values obtained from ratio ofthe strains in addition to given reference values.). In these cases, theinitial values utilized for solving the equations from (143) to (145)can be determined with various interporation method (quadratureinterporation, cosine interporation, Lagrange's interporation, splineinterporation) such that the values are spatially continuous with thereference values and the initial values determined from the a prioriinformation. The threshold A being dependent on the power or theaccuracy (SNR) of the strain data at each point, the threshold can bespatio-temporally changeable. The threshold can be set as small valuewhen or where the SNR of the strain is high, while the threshold can beset as large value when or where the SNR of the strain is low.Otherwise, in the unknown points or the unknown regions where therelative shear modulus values obtained from stain ratio with respect tothe reference values are larger than the relative value B (threshold),in a similar way, the shear modulus can be stably estimated using theabsolute reference values or the relative reference values (referencevalues obtained from ratio of the strains in addition to given referencevalues.). Also in this case, the initial values can be determined withvarious interporation method such that the values are spatiallycontinuous with the reference values and the initial values determinedfrom the a priori information. The threshold B being dependent on thepower or the accuracy (SNR) of the strain data at each point, thethreshold can be spatio-temporally changeable. The threshold can be setas high value when or where the SNR of the strain is high, while thethreshold can be set as low value when or where the SNR of the strain islow. The strain distribution data to determine reference regions ismoving-averaged with the spatio-temporally changeable window. Otherwise,to properly set the reference regions (values) and the initial values,the initial values can be calculated with various interporation method(including linear interporation), and freely the reference values andinitial values can be spatio-temporally low pass filtered. However,given μ(A) is unchangeable. Also on other equations, the referenceregions should be widely set, in a similar way, the initial values, thesingular points or the singular regions, the unknown points or theunknown regions can be dealt with. The method to set reference regionsexplained here can also be adopted when the direct method is utilized.

When solving equations from (143) to (145) derived from equations from(125) to (137″) by the iterative method for each unknown elastic modulusdistribution and each unknown visco elastic modulus distribution, byproperly setting initial values of the estimates, calculation amount canbe reduced. For instance, when solving equation (135) or (136) forunknown shear modulus distribution, the initial values can be determinedfrom the above-described strain ratio. In the above-described singularpoints, the singular regions, the points or regions where the absolutestrain is less than the positive value A (threshold), or the points orregions where the relative shear modulus values obtained from stainratio with respect to the reference values are larger than the relativevalue B (threshold), the initial values can be determined with variousinterporation method (quadrature interporation, cosine interporation,Lagrange's interporation, spline interporation) such that the values arespatially continuous with the reference values and the initial values(the initial values determined from the a priori information or strainratio). Otherwise, in the above-described singular points, the singularregions, the points or regions where the absolute strain is less thanthe positive value A (threshold), or the points or regions where therelative shear modulus values obtained from stain ratio with respect tothe reference values are larger than the relative value B (threshold),to properly set the initial values, the initial values can be calculatedwith various interporation method (including linear interporation) fromthe reference values and the initial values (the initial valuesdetermined from the a priori information or strain ratio), and freelythe reference values and the initial values can be spatio-temporally lowpass filtered. However, given A (A) is unchangeable. These thresholdsbeing dependent on the power or the accuracy (SNR) of the strain data ateach point, these thresholds can be spatio-temporally changeable. Thethresholds A and B can respectively be set as small and high values whenor where the SNR of the strain is high, while the thresholds A and B canrespectively be set as large and low values when or where the SNR of thestrain is low. Regarding with other elastic modulus distributions orother visco elastic modulus distributions, in a similar way, the initialvalues can be dealt with.

Regarding with some elastic moduli and visco elastic moduli, asabove-explained the reference regions (reference values) and the initialestimates are set and utilized, and simultaneously other elastic moduliand visco elastic moduli can be dealt with.

During iterative estimation, if elastic modulus, visco elastic modulus,time delay, relaxation time, density are estimated as the values out ofthe a priori known ranges, they are corrected such that they aresatisfied with the a priori data. For instance, the (visco) elasticmoduli are positive values. The (visco) Poisson's ratio is less than0.5. Then, for instance, if the (visco) elastic moduli are estimated asnegative values, they are corrected as positive values but nearly equalsto zero. If the (visco) Poisson's ratio are estimated to be larger than0.5, they are corrected to be smaller than 0.5 but nearly equals to 0.5.If plane stress condition is assumed, the (visco) Poisson's ratio can becorrected to be 0.5.

On the 1D or 2D measurement of the elastic constants such as the shearmodulus, the Poisson's ratio, etc., and visco elastic constants such asthe visco shear modulus, the visco Poisson's ratio, etc., they areestimated to be smaller than the original values when the point ofinterest gets far from the mechanical source. In this case, the sameshape model having homogeneous elastic modulus and visco elastic modulusand the same mechanical source model are utilized, the analytically ornumerically estimated strain data and strain rate data can be utilizedto correct the measured strain data and strain rate data. Otherwise, onthis model analytically or numerically estimated stress data can beutilized to correct measured elastic modulus distribution and viscoelastic modulus distribution. Otherwise, on this model the elasticmodulus and visco elastic modulus are estimated from the analytically ornumerically estimated strain data and strain rate data, and theestimates can be utilized to correct measured elastic modulusdistribution and visco elastic modulus distribution.

The temporal absolute change of the elastic constants such as the shearmodulus, the Poisson's ratio, etc., visco elastic constants such as thevisco shear modulus, the visco Poisson's ratio, etc., time delay,relaxation time can be obtained as the difference of the estimatedabsolute values. The temporal relative change of the elastic constants,visco elastic constants, time delay, relaxation time can be obtained asthe ratio of the estimated absolute or relative values, or regardingwith the elastic constants or the visco elastic constants, the temporalrelative change can be obtained as the difference of the estimatedlogarithms of them. In this way, on signal processing regarding with theelastic constants or the visco elastic constants, the logarithm can beutilized.

When iteratively solving the equations from (143) to (145), the initialestimate can be obtained from previous time estimate; reducing thecalculation amount. During iterative estimation, if elastic modulus,visco elastic modulus, time delay, relaxation time, density areestimated as the values out of the a priori known ranges, they arecorrected such that they are satisfied with the a priori data. Forinstance, the (visco) elastic moduli are positive values. The (visco)Poisson's ratio is less than 0.5. Then, for instance, if the (visco)elastic moduli are estimated as negative values, they are corrected aspositive values but nearly equals to zero. If the (visco) Poisson'sratio are estimated to be larger than 0.5, they are corrected to besmaller than 0.5 but nearly equals to 0.5. If plane stress condition isassumed, the (visco) Poisson's ratio can be corrected to be 0.5.

The above-explained regularization parameter can be set larger valuewhen the point of interest gets far from the reference region alongdominant tissue deformation direction.

On equations from (125) to (137″), the spectrum of the unknown elasticconstants and unknown visco elastic constants are handled, where theregularization method can be applied not only in the above-describedspatio-temporal directions but also in the frequency direction.

For instance, in the 1D ROI (x axis), when measurement target arefrequency variance (spectrum component distribution and phasedistribution) of the sequence of shear modulus distribution μ(x,t) andthe sequence of visco shear modulus distribution μ′(x,t), the discretesequence μ(x,j) [j=t/Δt(=0˜n)] of μ(x,t) can be expressed as

${{\mu\left( {x,j} \right)} = {\frac{1}{n + 1}{\sum\limits_{l = 0}^{n}{\begin{bmatrix}{\mu\left( {x,l} \right)} \\{\exp\left( {j\;{\theta_{\mu}\left( {x,l} \right)}} \right)}\end{bmatrix}\begin{bmatrix}{{\cos\left( {2\pi\; l\;\Delta\; f\; j\;\Delta\; t} \right)} +} \\{j\;{\sin\left( {2\pi\; l\;\Delta\; f\; j\;\Delta\; t} \right)}}\end{bmatrix}}}}}\;$where μ(x,l) and θ_(μ)(x,l) are respectively the spectrum component ofthe frequency l and the phase of the frequency l. j expresses imaginaryunit. l(=0˜n) is the discrete frequency coordinate (f=lΔf).

The discrete sequence μ′(x,j) [j=t/Δt(=0˜n)] of μ′(x,t) can be expressedas

${{\mu^{\prime}\left( {x,j} \right)} = {\frac{1}{n + 1}{\sum\limits_{l = 0}^{n}{\begin{bmatrix}{\mu^{\prime}\left( {x,1} \right)} \\{\exp\left( {j\;{\theta_{\mu^{\prime}}\left( {x,1} \right)}} \right)}\end{bmatrix}\begin{bmatrix}{{\cos\left( {2\pi\; 1\Delta\; f\; j\;\Delta\; t} \right)} +} \\{j\;{\sin\left( {2\pi\; 1\;\Delta\; f\; j\;\Delta\; t} \right)}}\end{bmatrix}}}}}\;$where μ′(x,l) and θ_(μ′)(x,l) are respectively the spectrum component ofthe frequency l and the phase of the frequency l.

Then, the first order differential equation (137) can be expressed as

$\begin{matrix}{{\sum\limits_{l = 0}^{n}{\left\lbrack {{\begin{pmatrix}\begin{matrix}{ɛ_{x\; x}\left( {x,j} \right)} \\{{\frac{\partial}{\partial x}{\mu\left( {x,l} \right)}} +} \\{\frac{\partial}{\partial x}{ɛ_{{x\; x}\;}\left( {x,j} \right)}}\end{matrix} \\{\mu\left( {x,l} \right)}\end{pmatrix}{\exp\begin{pmatrix}{j\;\theta_{\mu}} \\\left( {x,l} \right)\end{pmatrix}}} + {\begin{pmatrix}\begin{matrix}{ɛ_{x\; x}^{\prime}\left( {x,j} \right)} \\{{\frac{\partial}{\partial x}{\mu^{\prime}\left( {x,l} \right)}} +} \\{\frac{\partial}{\partial x}{ɛ_{x\; x}^{\prime}\left( {x,j} \right)}}\end{matrix} \\{\mu^{\prime}\left( {x,l} \right)}\end{pmatrix}\exp\;\left( {j\;{\theta_{\mu^{\prime}}\left( {x,l} \right)}} \right)}} \right\rbrack\left\{ {{\cos\left( {2\pi\; l\;\Delta\mspace{11mu} f\; j\;\Delta\; t} \right)} + {j\;{\sin\left( {2\pi\; l\;\Delta\; f\; j\;\Delta\; t} \right)}}} \right\}}} = 0} & (146)\end{matrix}$

Thus, with respect to each frequency l, the following simultaneous firstorder differential equations hold.

$\begin{matrix}{{{\left( {{{ɛ_{x\; x}\left( {x,j} \right)}\frac{\partial}{\partial x}{\mu\left( {x,l} \right)}} + {\frac{\partial}{\partial x}{ɛ_{x\; x}\left( {x,j} \right)}{\mu\left( {x,l} \right)}}} \right){\cos\left( {\theta_{\mu}\left( {x,l} \right)} \right)}} + {\left( {{{ɛ_{x\; x}^{\prime}\left( {x,j} \right)}\frac{\partial}{\partial x}{\mu^{\prime}\left( {x,l} \right)}} + {\frac{\partial}{\partial x}{ɛ_{x\; x}^{\prime}\left( {x,j} \right)}{\mu^{\prime}\left( {x,l} \right)}}} \right){\cos\left( {\theta_{\mu^{\prime}}\left( {x,l} \right)} \right)}}} = 0} & (146) \\{{{\left( {{{ɛ_{x\; x}\left( {x,j} \right)}\frac{\partial}{\partial x}{\mu\left( {x,l} \right)}} + {\frac{\partial}{\partial x}{ɛ_{x\; x}\left( {x,j} \right)}{\mu\left( {x,l} \right)}}} \right){\sin\left( {\theta_{\mu}\left( {x,l} \right)} \right)}} + {\left( {{{ɛ_{x\; x}^{\prime}\left( {x,j} \right)}\frac{\partial}{\partial x}{\mu^{\prime}\left( {x,l} \right)}} + {\frac{\partial}{\partial x}{ɛ_{x\; x}^{\prime}\left( {x,j} \right)}{\mu^{\prime}\left( {x,l} \right)}}} \right){\sin\left( {\theta_{\mu^{\prime}}\left( {x,l} \right)} \right)}}} = 0} & \left( 146^{\prime} \right)\end{matrix}$

The simultaneous differential equations (146′) and (146″) can be finitedifference approximated or finite element approximated in the same waywhere the equation (137) is dealt with at each time j(=0˜n).

By substituting the known nodal distribution of the real components andimaginary components of the spectrum of the frequency l(=0˜n) of theelastic constant and visco elastic constant [μ(I,l) cos θ_(μ)(I,l),μ(I,l) sin θ_(μ)(I,l), μ′(I,l) cos θ_(μ′)(I,l), μ′(I,l) sinθ_(μ′)(I,l)], at each time j (j=0˜n), simultaneous equations (142) arederived each for real components μ(I,l) cos θ_(μ)(I,l) and μ′(I,l) cosθ_(μ′)(I,l), and imaginary components μ(I,l) sin θ_(μ)(I,l) and μ′(I,l)sin θ_(μ′)(I,l).

In this way, on equations from (125) to (137″), the simultaneousequations are derived respectively for real components of the spectrumof the elastic constants and visco elastic constants, and imaginarycomponents of the spectrum of the elastic constants and visco elasticconstants. When respective simultaneous equations are regularized, asabove-explained, usually, for instance, the derived algebraic equationsare normalized by the root square of the summation of the powers of thespatially inhomogeneous coefficient distributions multiplied to theunknown distributions.

(A) Two equations derived on each sequence i(=1˜M), each time j(=0˜n),each frequency l(=0˜n), are respectively solved for of the frequency lreal component distributions and imaginary component distributions ofthe spectrum of the unknown parameters.

(B) Respective two equations derived from different sequence i(=1˜M),different time j(=0˜n), are simultaneously set for of the frequency lreal component distributions and imaginary component distributions ofthe spectrum of the unknown parameters, and solved.

(C) Respective two equations derived from different sequence i(=1˜M),different time j(=0˜n), are simultaneously set for of the frequency lreal component distributions and imaginary component distributions ofthe spectrum of the unknown parameters, and by spatial regularizationstably solved.

(D) Respective two equations derived from different sequencei(=1˜M),different time j(=0˜n), are simultaneously set for of thefrequency l real component distributions and imaginary componentdistributions of the spectrum of the unknown parameters, and by temporalregularization stably solved.

(E) Respective two equations derived on each sequence i(=1˜M), each timej(=0˜n), each frequency l(=0˜n), are simultaneously set for realcomponent distributions and imaginary component distributions of thespectrum of the unknown parameters, and by frequency regularizationstably solved. Spatial, and temporal regularization can alsosimultaneously be performed.

As above-explained, by one of (A), (B), (C), (D), (E), the frequencyvariances of the unknown elastic constants and visco elastic constantscan be obtained.

The sequences of the nodal elastic constant distributions and nodalvisco elastic constant distributions can be obtained by inverseFourier's transform of the spectrums. For instance, the sequence of thenodal shear modulus distribution is

${{\mu\left( {I,j} \right)} = {\frac{1}{n + 1}{\sum\limits_{j = 0}^{n}{\begin{bmatrix}{\mu\left( {I,1} \right)\exp} \\\left( {j\;{\theta_{\mu}\left( {I,j} \right)}} \right)\end{bmatrix}\begin{bmatrix}{{\cos\left( {2{\pi 1\Delta}\; f\; j\;\Delta\; t} \right)} +} \\{j\;{\sin\left( {2{\pi 1\Delta}\; f\; j\;\Delta\; t} \right)}}\end{bmatrix}}}}},$from which the sequence of the shear modulus distribution μ(x,t) can beobtained.

On also equations from (125) to (137″), the sequences of the nodalelastic constant distributions and nodal visco elastic constantdistributions can be obtained by inverse Fourier's transform of thespectrums.

The deformation fields are measured with changing the frequency of themechanical source, or by utilizing broadband mechanical source.

When instantaneous frequency of the deformation data can be measured,the frequency l can be dealt with as the instantaneous frequency.

Fourier's transform can be applied not only for time direction but alsospatial direction.

On equations (126), (127), (128), (129), (131), (132), (133), (134),(136), (137) and (128′″), (128″″), (129′″), (129″″), (133′″), (133″″),(134′″), (134″″), (137′), (137″) in order to deal with frequencyvariances of the sequences of the elastic constants and visco elasticconstants, (126), (127), (128), (128″″), (129), (129″″), (131), (132),(133), (133″″), (134), (134″″), (136), (137), (137″) can be approximatedutilizing convolute integration as like (128′″), (129′″), (133′″),(134′″), (137′). For instance, equation (137) can be approximated as

$\begin{matrix}{{{\left\lbrack {\int_{t^{\prime}}^{t}{{\mu\left( {t - \tau} \right)}{ɛ_{11}^{\prime}(\tau)}\ {\mathbb{d}\tau}}} \right\rbrack_{,1} + \left\lbrack {\int_{t^{\prime}}^{t}{{\mu^{\prime}\left( {t - \tau} \right)}{ɛ_{11}^{''}(\tau)}\ {\mathbb{d}\tau}}} \right\rbrack_{,1}} = 0},} & \left( 137^{\prime\prime\prime} \right)\end{matrix}$where t′ is initial time, ε″₁₁(t) is first order derivative of thestrain rate ε′₁₁(t).

As like on (128′″), (129′″), (133′″), (134′″), (137′), regularizationcan be performed temporally and spatially.

After Fourier's transform, regularization can also be performed spatialdirection, time direction, and in frequency domain. For instance,equation (137′″):[N(I,l)E′ ₁₁(I,l)]_(,1) +[N′(^(I,l))E″ ₁₁(I,l)]_(,1)=0,where E′₁₁(I,l) is Fourier's transform of the strain rate ε′₁₁(I,j), andE″₁₁(I,l) is Fourier's transform of the first order derivative of thestrain rate ε″₁₁(I,j). From Fourier's transform E₁₁(I,l) and E′₁₁(I,l)respective of the strain ε(I,j) and the strain rate ε′₁₁(I,j), E′₁₁(I,l)and E″₁₁(I,l) can be obtained as

$\begin{matrix}\begin{matrix}{{E_{11}^{\prime}\left( {x,1} \right)} = {\left( {j\; 2\;\pi\; l\;\Delta\; f} \right){E_{11}\left( {x,l} \right)}}} \\{{E_{11}^{\prime\prime}\left( {x,1} \right)} = {\left( {j\; 2\;\pi\; l\;\Delta\; f} \right){E_{11}^{\prime}\left( {x,1} \right)}}} \\{= {\left( {j\; 2\;\pi\; l\;\Delta\; f} \right)^{2}{E_{11}\left( {x,1} \right)}}}\end{matrix} & \left( 137^{\prime\prime\prime\prime} \right)\end{matrix}$

When dealing with density, density can also be obtained throughregularization.

In order to determine the unknown elastic constant distributions, theunknown visco elastic constant distributions, the unknown densitydistribution, equations from (125) to (137″) can also be solvedutilizing elastic constant data, visco elastic constant data, densitydata obtained on equations from (125) to (137″), and other deformationdata.

Next, utilizing the flowchart of FIG. 26, explained is measurementprocedure of the elastic constant distributions such as the shearmodulus, the Poisson's ratio, etc., the visco elastic constantdistributions such as the visco shear modulus, the visco Poisson'sratio, etc., time delay distributions, relaxation time distributions,and the density distributions. At first, reference regions are properlyset for the unknown elastic constants, the unknown visco elasticconstants, the unknown density (S11). Otherwise, as the referenceregion, reference points are set in the ROI 7. Reference point has knownelastic constants, known visco elastic constants, or known density.Otherwise, the reference point has the reference unity value, or otherfinite reference values.

To obtain high accuracy the elastic constants, the visco elasticconstants, and the density, the reference regions should be set suchthat they should widely cross the direction of the dominant tissuedeformation. The reference region has known elastic constantdistributions, known visco elastic constant distributions, known densitydistribution, or a priori assumed distributions. When measuring theabsolute elastic constant distributions, the absolute visco elasticconstant distributions, the absolute density distribution, the givenreference values must be absolute values.

Occasionally, by assuming the stress distribution in the referenceregion, from measured strain values the reference elastic constant isdetermined. (For instance, by assuming the stress distribution to beconstant, from strain ratio the reference elastic constant can bedetermined.). Moreover, by assuming the stress distribution in thereference region, from strain rate values the reference visco elasticconstant is determined. (For instance, by assuming the stressdistribution to be constant, from strain rate ratio the reference viscoelastic constant can be determined.).

When there exist neither reference point nor reference regions, ifreference medium can directly be contacted to object, the deformations(strain tensor field, strain rate tensor field, acceleration vectorfield) are measured over the ROI including the reference (S12). In thiscase, the shear modulus value of the reference should be large comparedwith that of the target. The reference medium should be put between themechanical source 8 and the ROI.

As the object is deformed in 3D space, 3D reconstruction should becarried out. However, when estimating in the superficial tissues theelastic constants, the visco elastic constants, and the density, 1Dreconstruction method [from (135) to (137″)] is useful since utilizedcan be accurately measured strain data, strain rate data, andacceleration data in beam direction. In contrast, when estimating in thedeeply situated tissues the elastic constants, the visco elasticconstants, and the density, multi-dimensional reconstruction method isuseful since the freedom of configurations can be high of mechanicalsources and reference regions (mediums).

Specifically, on 2D reconstruction, when 2D strain distributionapproximately occurs, equations from (125) to (129″″) can be utilized.Alternatively, when 2D stress distribution approximately occurs,equations from (130) to (134″″) can be utilized. To measure independentdeformation fields (strain tensor fields, strain rate tensor fields,acceleration vector fields), the position of the mechanical source 8 ischanged. Since the measurement accuracy of the strains, strain rates,acceleration vectors rely on their magnitudes, to measure the elasticconstants, the visco elastic constants, the density with uniformaccuracy over the ROI, the position of the mechanical source 8 should bevariously changed. This measurement accuracy has the relationship oftrade off between the measurement time and the cost. As alreadydescribed, when the object is spontaneously deformed due to mechanicalsources 8′ and 8″, the mechanical source 8 may not be needed.

The measurement controller 3 controls the positions of the object 6 andthe displacement (strain) sensor 5, and the measurement controller 3inputs the position information and the echo signals into the storage 2.At the data processor 1, measured strain data, strain rate data,acceleration data are filtered to reduce noises (S13), by whichspatially smoothed coefficients E and e are obtained (S14).Subsequently, the elastic constant distributions, the visco elasticconstant distributions, the density distribution are obtained from thenormal equations (144) (S15). Thus, measurement results are, at eachtime, displacement vector distribution, strain tensor distribution,gradient distribution of the strain tensor, strain tensor ratedistribution, gradient distribution of the strain rate tensor, elasticconstant distributions such as shear modulus, Poisson's ratio, Lameconstants, etc., visco elastic constant distributions such as viscoshear modulus, visco Poisson's ratio, visco Lame constants, etc., timedelay distributions or relaxation time distributions relating theseelastic constants and visco elastic constants, density distribution,gradient distributions of these results, Laplacian distributions ofthese results, temporal first derivatives of these results, temporalsecond derivatives of these results. To store time series of thesemeasurement results, these measurement results (output of data processor1) are input into the storage 2. To display in real time thesemeasurement results on CRT (color or gray), the output of data processor1 can be input into display equipment. Freeze image can also bedisplayed. When displaying these measurement results, each measurementresult can be truncated by respectively set upper value or lower value.When displaying elastic constant distributions or visco elastic constantdistributions, the inversion can be also displayed. Moreover, directcurrent can be added to the measurement results, or subtracted from themeasurement results. When displaying strain tensor distribution, to makethe sign of the strain invariant, the direct current can be added (thebrightness should be assigned such that the strain image has correlationwith the elastic constant image). Furthermore, each measurement resultcan also be displayed in log scaled.

Measurement results are, at each time, displacement vector distribution,strain tensor distribution, gradient distribution of the strain tensor,strain tensor rate distribution, gradient distribution of the strainrate tensor, elastic constants such as shear modulus, Poisson's ratio,Lame constants, etc., visco elastic constants such as visco shearmodulus, visco Poisson's ratio, visco Lame constants, etc., time delaydistributions or relaxation time distributions relating these elasticconstants and visco elastic constants, density distribution, gradientdistributions of these results, Laplacian distributions of theseresults, temporal first derivatives of these results, temporal secondderivatives of these results, time series of these results, time seriesof relative (ratio) changes of these results or absolute (substraction)changes of these results, frequency variance distributions of theseresults, elastic energy at each time or accumulations, consumed energyat each time or accumulations, time series of elastic energy at eachtime or accumulations, consumed energy at each time or accumulations,time series of relative (ratio) changes of these energy or absolute(substraction) changes of these energy. When there exist no stain datapoint or no strain data region, the elastic constants etc. areinterporated or extraporated from measured ones. The results can bestored at storage 2, and can be displayed.

These results can be obtained through spatial filtering of the absoluteelastic constant distributions, absolute visco elastic constantdistributions, absolute time delay distributions, absolute relaxationtime distributions, absolute density distributions, or relative thesedistributions obtained from the normal equations (144). Otherwise,obtained these results can be spatially filtered. Otherwise, theseresults can be obtained through temporal or spatial or temporal-spatialfiltering of the elastic constant distributions, visco elastic constantdistributions, time delay distributions, relaxation time distributions,density distributions. Otherwise, obtained these results can betemporally or spatially or temporal-spatially filtered. These resultscan be stored at storage 2, and can be displayed. The spatial filter,the temporal filter, and the temporal-spatial filter are high pass type,band pass type, low pass type. These filters can be freely utilized atdata processor 1.

For equations (125) to (137″), the unknown elastic constantdistribution, the unknown visco elastic constant distribution, theunknown density distribution can be obtained from the measured elasticconstant distribution, visco elastic constant distribution, densitydistribution using another deformation field data, or obtained fromtheir typical value distributions.

By utilizing the ultrasonic diagnosis equipment together, the spatialvariations of the bulk modulus and the density can be measured together,and can be displayed together. In this case, utilized together are thedata processor 1, the data storage 2, the measurement controller 3,displacement (strain) sensor 5, transmitting/output controller 5′ etc(FIG. 1). By utilizing the magnetic nuclear imaging equipment together,the atomic density distribution can be measured together, and can bedisplayed together.

As above-described (FIG. 1), using the displacement (strain) sensor,remotely measured can be strain tensor field, strain rate tensor field,acceleration vector field. By solving by finite difference method orfinite element method the first order partial differential equationswhose coefficients are derived from the measured data, estimated can bethe absolute elastic constant distributions, the relative elasticconstant distributions with respect to the reference elastic constants,the absolute visco elastic constant distributions, the relative viscoelastic constant distributions with respect to reference elasticconstants, the absolute density distribution, the relative densitydistribution with respect to reference density.

By using the regularized algebraic equations, the errors (measurementnoises) of the measured strain data, strain rate data, acceleration datacan be coped with. Moreover, ill-conditioned reference regions can alsobe coped with.

Elasticity and visco-elasticity constants measurement apparatus, relatedto this conduct form is useful to monitor treatment effectiveness ofradiation therapy since degeneration and change of temperature has highcorrelation with the change of elastic constants such as shear modulus,Poisson's ratio, Lame constants, etc., visco elastic constants such asvisco shear modulus, visco Poisson's ratio, visco Lame constants, etc.,delay times or relaxation times relating these elastic constants andvisco elastic constants, and density.

On conduct form of FIG. 1, as an example, the ultrasound transducer isutilized as the displacement (strain) sensor 5 to measure strain tensor,strain rate tensor, acceleration vector. On the present invention,however, strain tensor, strain rate tensor, acceleration vector can bemeasured by signal processing of the magnetic nuclear resonance signals,and from these deformation data measured can be elastic constants suchas shear modulus, Poisson's ratio, Lame constants, etc., visco elasticconstants such as visco shear modulus, visco Poisson's ratio, visco Lameconstants, etc., delay times or relaxation times relating these elasticconstants and visco elastic constants, and density.

The next explanation is the treatment apparatus related to one ofconduct forms of the present invention. This treatment apparatus usesfor ultrasound therapy the above-explained measurement technique ofdisplacement vector field/strain tensor field, and measurement techniqueof elastic constants, visco elastic constants, and density.

The aim of the measurement of the followings is to quantitativelyexamine statically or dynamically the objects, substances, materials,living things, etc., i.e., displacement vector distribution, the straintensor distribution, the strain rate tensor distribution, theacceleration vector distribution, the velocity vector distribution,elastic constant distributions, visco elastic constant distributions.For instance, on human soft tissues, the tissues can be differentiatedby extra corporeally applying pressures or low frequency vibrations,namely, by focusing on the change of the elasticity due to growth oflesion or change of pathological state. Instead of the pressures and thevibrations spontaneous hear motion or pulse can also be utilized tomeasure tissue deformation, and tissues can be differentiated from thevalues and distributions of tissue elastic constants and visco elasticconstants. Blood velocity can also be observed.

FIG. 27 shows the global structure of the treatment apparatus related tothis conduct form. On therapy field, lesions can be treated by applyinghigh intensity ultrasound, laser, electromagnetic RF wave,electromagnetic micro wave, or by cryotherapy. On these low invasivetreatments, degeneration occurs, composition rate in weight changes, andtemperature changes. For instance, on living tissues, proteindegenerates, and tissue coagulates. The degeneration, change ofcomposition rate, and change of temperature occur together with changesof elastic constants such as shear modulus, Poisson's ratio, etc., viscoelastic constants such as visco shear modulus, visco Poisson's ratio,etc., delay times or relaxation times relating these elastic constantsand visco elastic constants, or density.

Thus, by measuring lesion's absolute or relative shear modulus, absoluteor relative Poisson's ratio, absolute or relative visco shear modulus,absolute or relative visco Poisson's ratio, absolute or relative delaytimes or absolute or relative relaxation times, or absolute or relativedensity, etc., and by observing these time courses or these frequencyvariances, effectiveness of the treatments can be low invasivelymonitored. Based on conversion data for each tissue obtained fromtheories, simulations and measurements, changes of measured shearmodulus, Poisson's ratio, visco shear modulus, visco Poisson's ratio,delay time, relaxation time, density, strain, strain rate, can beconverted into consumed electric energy, time course of electric energy,temperature, or time course of temperature. From the measured consumedelectric energy, the time course of electric energy, the temperature, orthe time course of temperature, effectiveness of the treatment can beconfirmed.

The consumed electric energy, the time course of electric energy canalso be measured by using electric power meter and tissue physicalparameters (tissue electric impedance, mechanical impedance, etc.). Thetemperature, or the time course of temperature can also be measured byusing usual temperature monitoring method, thermo coupler, etc. Bymeasuring these spatial distributions, not only effectiveness oftreatment can be monitored, but also safety and reliability can beobtained. These monitoring data can be utilized for dynamic electronicdigital control or mechanical control of beam focus position, treatmentinterval, ultrasound beam power, ultrasound beam strength, transmitterm, transmit interval, beam shape (apodization), etc. Thus, thesemonitoring data can be utilized to improve the efficiency of thetreatment.

FIG. 27 shows the treatment apparatus which transmits high intensityultrasounds to lesion. The treatment apparatus can be equipped withultrasound diagnosis equipment and elasticity and visco-elasticityconstants measurement apparatus. As shown in FIG. 27, the treatmentprobe 11 possesses the ultrasound transducer 12 and treatment transducer13 (treatment transducer can also serve as the ultrasound transducer12), supporter 14. As utilized on the ultrasonic diagnosis equipment(for instance, convex type transducer), the ultrasound transducer 12arrays plural oscillators. The treatment transducer 13 also arraysplural oscillators. In the figure, concavity type treatment probe 11 isshown. The supporter 14 can be held by hand or position controller 4, bywhich the position of the treatment probe 11 can be controlled.

To the treatment transducer 13, the ultrasound pulse generated at thetreatment pulse generator 21 is provided through the treatment wavedelay circuit 22 and amplifier 23. That is, at the treatment wave delaycircuit 22 the delay time of the transmit ultrasound pulse is controlledfor each oscillator, by which the focus position of the synthesizedultrasound beam is controllable.

To the oscillators of the ultrasound transducer 12, the ultrasoundpulses generated at the ultrasound pulse generator 31 are providedthrough the transmit and receive separator 34 after being focused at thetransmit delay circuit and being amplified at the amplifier 33. The echosignals received at the oscillators of the ultrasound transducer 12 areamplified at the amplifier 35 after passing through the transmit andreceive separator, and the phases of the echo signals are matched at thephase matcher 36. The outputs of the phase matcher 36 are used toreconstruct image at the signal processor 37, and the image data isconverted to diagnosis image at the DSC (digital scan converter) 38, andthe diagnosis image is displayed at the monitor 39. Known ultrasounddiagnosis equipment can be used for this diagnosis equipment.

The elastic constants and visco elastic constants measurement part 40related to this conduct form can measure shear modulus, Poisson's ratio,visco shear modulus, visco Poisson's ratio, density, delay times orrelaxation times relating these elastic constants and visco elasticconstants, density, etc., using the echo signals output from phasematcher 36. The measured data and calculated results are stored at thedata storage equipped with 40.

Commands of the controller 41 control the treatment pulse generator 21,the treatment wave delay circuit 22, the ultrasound pulse generator 31,the transmit delay circuit 32, the phase matcher 36, the signalprocessor 37, the DSC 38, and the elastic constants and visco elasticconstants measurement part 40. The operator can input commands andconditions from the operation part 42 into the controller 41, by whichthe operator can set various operation conditions and treatmentconditions. The signal processor 37, elastic constants and visco elasticconstants measurement part 40, operation part 42, controller 41 arecomprised of computers.

Next explanation is how this like ultrasound treatment equipment isutilized. The treatment probe 11 is contacted onto body surface, and issupported such that the ROI include the target lesion. Occasionally, byusing water tank, the treatment probe 11 is supported without contactingonto the body surface. At first, to image the lesion part, the commandto start imaging is input from the operation part 42, by which as theresponse the controller 41 outputs the commands to the ultrasound pulsegenerator 31 and the transmit delay circuit 32. Then, the ultrasoundbeam is transmitted from the ultrasound transducer 12. This ultrasoundbeam scans the ROI. The echo signals are received at the oscillators ofthe ultrasound transducer, the phases of the echo signals are matched atthe phase matcher 36. The outputs of the phase matcher 36 are used toreconstruct image at the signal processor 37, and the image data isconverted to diagnosis 2D image at the DSC (digital scan converter) 38,and the diagnosis image is displayed at the monitor 39. Thus, duringobserving the images and diagnosing tissues, when the lesion part can bedetected, treatment is carried out.

That is, when the lesion is detected, the treatment probe is held at thepresent position. From the image memorized at the DSC 38, the controller41 obtains the delay time to provide the drive pulse to each oscillatorof the treatment transducer. Then, the controller outputs the obtainedtime delays into the treatment wave delay circuit 22, by which thelesion part is focused. The strength of the ultrasound beam can becontrolled. The lesion part is heated. The lesion part degenerates. Thetreatment can also be carried out by observing 3D ultrasound image.Controlled of treatment ultrasound beam can be beam focus position,treatment interval, ultrasound beam power, ultrasound beam strength,transmit term, transmit interval, beam shape (apodization), etc.

Next explanation is the procedure of treatment and measurement of shearmodulus, Poisson's ratio, visco shear modulus, visco Poisson's ratio,time delay, relaxation time density, etc. for monitoring the treatmenteffectiveness. Flowchart of FIG. 28 is referred to. At first, before thetreatment, measured in the ROI are shear modulus distribution μ(x,y,z),Poisson's ratio ν(x,y,z), visco shear modulus μ′(x,y,z), visco Poisson'sratio ν′(x,y,z), delay time τ (x,y,z), relaxation time τ′(x,y,z),density ρ(x,y,z) (S21). Command is sent from the operator part 42 to thecontroller 41, after which the ultrasounds are transmitted from theultrasound transducer 12. Subsequently, the controller 41 sends commandto the elastic constants and visco elastic constants measurement part40, by which using echo signals output from the phase matcher 36 thestrain tensor field or strain rate tensor field are measured. From themeasured strain tensor field or strain rate tensor field, calculated areshear modulus distribution μ(x,y,z), Poisson's ratio ν(x,y,z), viscoshear modulus μ′(x,y,z), visco Poisson's ratio ν′(x,y,z), delay timeτ(x,y,z), relaxation time τ′(x,y,z), density ρ(x,y,z), etc.

Next, if the lesion part is confirmed, the treatment process counter Iis initialized (I=0) (S22). The starting position of the treatment andthe initial strength of the treatment ultrasound are set (S23), and thetreatment is started (S24). At every treatment, measured are shearmodulus distribution μ(x,y,z), Poisson's ratio distribution ν(x,y,z),visco shear modulus distribution μ′(x,y,z), visco Poisson's ratiodistribution ν′(x,y,z), delay time distribution τ(x,y,z), relaxationtime distribution τ′(x,y,z), density distribution ρ(x,y,z), etc. (S25).The measured elastic constants, visco elastic constants, delay times,relaxation times can be absolute values or spatially relative values ortemporally relative values. Then, to confirm the effectiveness of thetreatment, comparison can be carried out between shear modulus valueμ(x,y, z), Poisson's ratio value ν(x,y, z), visco shear modulus valueμ′(x,y, z), visco Poisson's ratio value ν′(x,y,z), etc., and theirrespective thresholds TH1 (softened case) and TH2 (hardened case), etc.(S26). Moreover, comparison can be carried out between delay time valueτ(x,y,z), relaxation time value τ′(x,y,z), density value ρ(x,y,z) andtheir respective thresholds. The thresholds TH1, TH2, etc., can be setfrom the information of the tissue properties etc. The thresholds TH1,TH2, etc. are the functions of the time t, the position (x,y,z),ultrasound parameters such as shooting counter etc., degenerationinformation, etc. The thresholds can be set before the treatment, or canbe updated during the treatment. If desired effectiveness cannot beconfirmed, the ultrasound strength is controlled to be higher (S27),after which the treatment is carried out again (S24). If the desiredeffectiveness can be confirmed, it is judged if the treatments of allthe positions are finished (S28). If the treatments of all the positionsare not finished yet, the treatment position is changed (S29), and thetreatment is carried out again (S24).

If the treatments of all the positions are finished, the treated part isnaturally or compulsively cooled down (S30). After the treatment,measured are shear modulus distribution μ(x,y,z), Poisson's ratiodistribution ν(x,y,z), visco shear modulus distribution μ′(x,y,z), viscoPoisson's ratio distribution ν′(x,y,z), delay time distributionτ(x,y,z), relaxation time distribution τ′(x,y,z), density distributionρ(x,y,z), etc. (S31). It is judged if desired effectiveness can beobtained at all the positions (S32). If the desired effectiveness cannot be confirmed at all the positions, till the effectiveness can beconfirmed, the treated part is cooled down (from S30 to S32). If thedesired effectiveness is confirmed at all the positions, it is judged ifthis treatment process is finished (S33). When the treatment process isnot finished, the treatment process counter I is incremented, and stepsfrom S23 to S33 are iteratively carried out. The maximum number of thetreatment process can be set. The treatment position can be set in orderfrom deep position or peripheral position, or the treatment position canbe set where the treatment effectiveness is not confirmed.

As described above, using the treatment apparatus of FIG. 27, during theultrasound treatment, we can observe the treatment effectiveness in realtime and then we can properly carry out the treatment. Moreover, byconfirming the treatment effectiveness in real time, the ultrasoundstrength, the shoot number, etc. can be controlled.

The treatment apparatus of FIG. 27 can also be used for the othertreatments such as laser treatment, electromagnetic RF wave treatment,electromagnetic micro wave treatment, or cryotherapy. In this case, thelow invasive treatment modalities are substituted for the treatmentprobe 11, the treatment pulse generator 21, the treatment wave delaycircuit 22, the amplifier 23.

As the ultrasound transducer 12, for instance, utilized can be 2D arrayaperture type applicator, 1D array aperture type applicator, concavitytype applicator. For instance, when carrying out cryotherapy or radiotherapeutics (applying high intensity focus ultrasound, laser,electromagnetic RF wave, microwave, etc.) on living things or the invitro tissues through skin, mouth, vagina, anus, opened body, bodysurface, monitored can be degeneration, change of composition rate inweight, and change of temperature. Measured shear modulus, Poisson'sratio, visco shear modulus, visco Poisson's ratio delay time, relaxationtime, density, etc., can be utilized as index to dynamically controlbeam position (focus), treatment interval, beam power, beam strength,transmit term, transmit interval, beam shape (apodization), etc.

Before, during, after the treatment, the followings can be displayed onmonitor 39 as static or motion or time course (difference) image, thevalues of arbitrary points, the time course (graph), etc., i.e., notonly elastic constant distribution or visco elastic constant but alsodisplacement vector distribution, displacement vector componentdistributions, strain tensor component distributions, strain gradientcomponent distributions, strain rate tensor component distributions,strain rate gradient component, acceleration vector componentdistributions, or velocity vector component distributions, etc.

Moreover, by utilizing ultrasound diagnosis apparatus together, spatialvariations of bulk modulus and density of tissues can be measured anddisplayed in real-time. On the ultrasound image, as measurement results,superimposed and displayed can be static or motion or time course(difference) images of the displacement vector distribution,displacement vector component distributions, strain tensor componentdistributions, strain gradient component distributions, strain ratetensor component distributions, strain rate gradient component,acceleration vector component distributions, velocity vector componentdistributions, etc.

Particularly when the applicator has an arrayed aperture, beam focusposition, treatment interval, beam power, beam strength, transmit term,transmit interval, beam shape (apodization), etc. are electronicallydigital controlled, while when the applicator has a concavity aperture,the focus position is mechanically controlled. The flowchart of FIG. 28can be applied to the control program, for instance. That is, to controlbeam focus position, treatment interval, beam power, beam strength,transmit term, transmit interval, beam shape (apodization), etc,utilized can be absolute or relative shear modulus distribution,absolute or relative Poisson's ratio distribution, absolute or relativevisco shear modulus distribution, absolute or relative visco Poisson'sratio distribution, absolute or relative delay time distributions,absolute or relative relaxation time distributions, absolute or relativedensity distribution, temporally absolute or relative changes of theseelastic constants, visco elastic constants, delay times, relaxationtimes, density, etc. measured before, during, after transmitting theenergies.

The above-explained measurement technique of displacement vector field,strain tensor field, etc., and measurement technique of elasticconstants, visco elastic constants, density, etc., can be utilizedtogether with interstitial needle, catheter, etc. when carrying outcryotherapy or radio therapeutics (applying high intensity focusultrasound, laser, electromagnetic RF wave, micro wave, etc.) or whennon-destructive examining living things or substances or materials(cases included during producing or growing.).

For instance, on interstitial cryotherapy, interstitial radiotherapeutics (applying high intensity focus ultrasound, laser,electromagnetic RF wave, micro wave, etc. utilizing needles and plate,only needles, mono needle, etc), etc., the followings can also bedisplayed on monitor before, during, after the treatment as static ormotion or time course (difference image) image, the values of arbitrarypoints, the time course (graph), etc., i.e., not only elastic constantdistribution or visco elastic constant but also displacement vectordistribution, displacement vector component distributions, strain tensorcomponent distributions, strain gradient component distributions, strainrate tensor component distributions, strain rate gradient component,acceleration vector component distributions, or velocity vectorcomponent distributions, etc. Moreover, by utilizing ultrasounddiagnosis apparatus together, spatial variations of bulk modulus anddensity of tissues can also be measured and displayed in real-time. Onthe ultrasound image, as measurement results, superimposed and displayedcan also be static or motion or time course (difference) images of thedisplacement vector distribution, displacement vector componentdistributions, strain tensor component distributions, strain gradientcomponent distributions, strain rate tensor component distributions,strain rate gradient component, acceleration vector componentdistributions, velocity vector component distributions, etc. Thefollowings can be displayed in vector style as well, i.e., thedisplacement vector distribution, acceleration vector, velocity vector.

To obtain safety when carrying out treatment, by setting the uppervalues and lower values of shear modulus, Poisson's ratio, visco shearmodulus, visco Poisson's ratio, delay times, relaxation times, density,etc., and by setting the upper values of absolute or relative changes ofthese, beam position (focus), treatment interval, beam power, beamstrength, transmit term, transmit interval, beam shape (apodization),etc. should be controlled such that these physical parameter values donot change more than necessary.

The treatment effectiveness can also be evaluated by measuringtemperature and temporal change of temperature as above-explained fromstrain (tensor) distribution, strain rate (tensor) distribution, shearmodulus distribution, Poisson's ratio distribution, visco shear modulusdistribution, visco Poisson's ratio distribution, density distribution,temporal changes of these, etc. measured before, during, aftertransmitting the energies. In this case, to obtain safety, by settingthe upper values of temperature or change of temperature, beam position(focus), treatment interval, beam power, beam strength, transmit term,transmit interval, beam shape (apodization), etc. should be controlledsuch that the temperature do not heighten more than necessary. These canalso be controlled utilizing shear modulus value μ, Poisson's ratiovalue ν, visco shear modulus value, visco Poisson's ratio value, densityvalue, delay time values, relaxation time values, strain values, strainrate values, etc. converted from the upper values. Temperature andchange of temperature can also be measured utilizing the conventionaltemperature measurement method or thermo coupler.

In cases where no mechanical source exists, or mechanical sources arenot utilized, degeneration, change of composition rate in weight, andchange of temperature can also be detected from strain (tensor)distribution, strain rate (tensor) distribution, shear modulusdistribution, Poisson's ratio distribution, visco shear modulusdistribution, visco Poisson's ratio distribution, density distribution,temporal changes of these, etc. measured before, during, aftertransmitting the energies. Directly the expansion and shrink can also bedetected when strain (tensor) distribution or strain rate (tensor)distribution are measured.

The elasticity and visco-elasticity constants measurement apparatus ofthe present invention can be utilized to monitor degeneration, change ofcomposition rate in weight, change of temperature due to injection ofmedicine, putting of medicine, giving of medicine. To control amount themedicine, term, interval, position, etc., utilized can be absolute orrelative shear modulus distribution, absolute or relative Poisson'sratio distribution, absolute or relative visco shear modulusdistribution, absolute or relative visco Poisson's ratio distribution,absolute or relative delay time distributions, absolute or relativerelaxation time distributions, absolute or relative densitydistribution, temporally absolute or relative changes of these elasticconstants, visco elastic constants, delay times, relaxation times,density, etc. measured before, during, after the treatment. Anticancerdrug can be utilized as the medicine.

That is, to monitor the treatment effectiveness (including change oftemperature) of anticancer drug and to control the treatment, thefollowings can also be displayed on monitor before, during, after thetreatment as static or motion or time course (difference) image, thevalues of arbitrary points, the time course (graph), etc., i.e., notonly elastic constant distribution or visco elastic constant but alsodisplacement vector distribution, displacement vector componentdistributions, strain tensor component distributions, strain gradientcomponent distributions, strain rate tensor component distributions,strain rate gradient component, acceleration vector componentdistributions, or velocity vector component distributions, etc.Moreover, by utilizing ultrasound diagnosis apparatus together, spatialvariations of bulk modulus and density of tissues can also be measuredand displayed in real-time. On the ultrasound image, as measurementresults, superimposed and displayed can also be static or motion or timecourse (difference) images of the displacement vector distribution,displacement vector component distributions, strain tensor componentdistributions, strain gradient component distributions, strain ratetensor component distributions, strain rate gradient component,acceleration vector component distributions, velocity vector componentdistributions, etc. The followings can be displayed in vector style aswell, i.e., the displacement vector distribution, acceleration vector,velocity vector. In cases where no mechanical source exists, ormechanical sources are not utilized, degeneration, expansion or shrink,and change of temperature, etc. can also be detected from displacementvector, strain (tensor) distribution, strain rate (tensor) distribution,etc.

The elastic constants, visco elastic constants, density, high order dataexpressed from elastic constants, visco elastic constants, density areutilized to obtain non-linear properties of tissues by linearapproximation of non-linear phenomena in infinitesimal time space orspatial space. Thus, estimated non-linear elastic constants data,non-linear visco elastic constants data, high order data expressed fromnon-linear data can be utilized for diagnosis and treatment.

Thus, as explained above, the present invention can realize accuratemeasurement in 3D space of interest (SOI) or 2D region of interest (ROI)or 1D ROI of displacement vector distribution, strain tensordistribution, the spatio or temporal derivatives of these, generated dueto arbitrary mechanical sources. If the target naturally deforms,elastic constant or visco elastic constant can be estimated in the SOIor ROI without disturbing the deformation field from measureddeformation data. Moreover, even if there exist another mechanicalsources and uncontrollable mechanical sources in the object, forinstance, the elastic constant and visco elastic constant measurementapparatus can be utilized, which is applicable for diagnosing the partof interest in the object and for monitoring the treatmenteffectiveness. Furthermore, low-invasive treatment apparatus can berealized, which is equipped with such elastic constant and visco elasticconstant measurement apparatus.

1. An elasticity and visco-elasticity constants measurement apparatuscomprising: storage means for storing at least one of strain tensordata, strain rate tensor data and acceleration vector data measured in aROI (reagion of interest) set in a target; and calculating means forcalculating at least one of elastic constants, visco elastic constantsand density of an arbitrary point within the ROI on the basis of atleast one of the measured strain tensor data, strain rate tensor dataand acceleration vector data; wherein the calculating means numericallyobtains at least one of the elastic constants, visco elastic constantsand density on the basis of at least one of a ratio of strains, a ratioof strain rates, and first order partial differential equationsrepresenting a relation between (i) at least one of the elasticconstants, visco elastic constants and density and (ii) at least one ofthe strain tensor data, strain rate tensor data and acceleration vectordata.
 2. The elasticity and visco-elasticity constants measurementapparatus according to claim 1, further comprising: output means foroutputting degeneration information based on at least one of thecalculated elastic constants, visco elastic constants, density, strains,strain rates and accelerations.
 3. The elasticity and visco-elasticityconstants measurement apparatus according to claim 1, furthercomprising: output means for outputting degeneration information bycomparing at least one of the calculated elastic constants, viscoelastic constants, density, strains and strain rates with those valuesset in advance.
 4. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 1, wherein said calculatingmeans utilizes at least one of the elastic constants, visco elasticconstants and density, which are measured in advance or set in advance,as coefficients of said at least one of ratio of strains, ratio ofstrain rates and first order partial differential equations.
 5. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 2, wherein said calculating means utilizes at leastone of the elastic constants, visco elastic constants and density, whichare measured in advance or set in advance, as coefficients of said atleast one of ratio of strains, ratio of strain rates and first orderpartial differential equations.
 6. The elasticity and visco-elasticityconstants measurement apparatus according to claim 3, wherein saidcalculating means utilizes at least one of the elastic constants, viscoelastic constants and density, which are measured in advance or set inadvance, as coefficients of said at least one of ratio of strains, ratioof strain rates and first order partial differential equations.
 7. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 1, wherein said calculating means utilizes at leastone of reference elastic constants, reference visco elastic constantsand reference density, which are measured in advance or set in advance,as initial conditions of said at least one of ratio of strains, ratio ofstrain rates and first order partial differential equations.
 8. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 2, wherein said calculating means utilizes at leastone of reference elastic constants, reference visco elastic constantsand reference density, which are measured in advance or set in advance,as initial conditions of said at least one of ratio of strains, ratio ofstrain rates and first order partial differential equations.
 9. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 3, wherein said calculating means utilizes at leastone of reference elastic constants, reference visco elastic constantsand reference density, which are measured in advance or set in advance,as the initial conditions of said at least one of ratio of strains,ratio of strain rates and first order partial differential equations.10. The elasticity and visco-elasticity constants measurement apparatusaccording to claim 4, wherein said calculating means utilizes at leastone of reference elastic constants, reference visco elastic constantsand reference density, which are measured in advance or set in advance,as initial conditions of said at least one of ratio of strains, ratio ofstrain rates and first order partial differential equations.
 11. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 5, wherein said calculating means utilizes at leastone of reference elastic constants, reference visco elastic constantsand reference density as initial conditions of said at least one ofratio of strains, ratio of strain rates and first order partialdifferential equations.
 12. The elasticity and visco-elasticityconstants measurement apparatus according to claim 6, wherein saidcalculating means utilizes at least one of reference elastic constants,reference visco elastic constants and reference density, which aremeasured in advance or set in advance, as initial conditions of said atleast one of ratio of strains, ratio of strain rates and first orderpartial differential equations.
 13. The elasticity and visco-elasticityconstants measurement apparatus according to claim 1, wherein saidcalculating means utilizes priori information during or aftercalculation.
 14. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 2, wherein said calculatingmeans utilizes priori information during or after calculation.
 15. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 3, wherein said calculating means utilizes prioriinformation during or after calculation.
 16. The elasticity andvisco-elasticity constants measurement apparatus according to claim 4,wherein said calculating means utilizes priori information during orafter calculation.
 17. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 5, wherein said calculatingmeans utilizes priori information during or after calculation.
 18. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 6, wherein said calculating means utilizes prioriinformation during or after calculation.
 19. The elasticity andvisco-elasticity constants measurement apparatus according to claim 7,wherein said calculating means utilizes priori information during orafter calculation.
 20. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 8, wherein said calculatingmeans utilizes priori information during or after calculation.
 21. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 9, wherein said calculating means utilizes prioriinformation during or after calculation.
 22. The elasticity andvisco-elasticity constants measurement apparatus according to claim 10,wherein said calculating means utilizes priori information during orafter calculation.
 23. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 11, wherein said calculatingmeans utilizes priori information during or after calculation.
 24. Theelasticity and visco-elasticity constants measurement apparatusaccording to claim 12, wherein said calculating means utilizes prioriinformation during or after calculation.
 25. The elasticity andvisco-elasticity constants measurement apparatus according to claim 1,further comprising at least one of: a controller of a means oftreatment, or scheme or protocol of treatment based on the measurements;and input means for inputting commands and conditions into thecontroller.
 26. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 1, further comprising: meansfor displaying as a static or motion image, color or gray image of atleast one of a displacement vector, displacement vector components, astrain tensor, strain tensor components, a strain rate tensor, strainrate tensor components, a velocity vector, velocity vector components,an acceleration vector, acceleration vector components, an elasticmodulus, a visco-elastic modulus, delay, relaxation time, density,elastic energy, consumed energy, accumulation of energy, gradient,Laplacian, a 1st temporal derivative, a 2nd temporal derivative,frequency variance, relative change, absolute change, truncated one byabove value or lower value, direct-current-added one,direct-current-subtracted one, reciprocal, log-scaled one, one of thesebeing superimposed on ultrasound image or NMR image.
 27. The elasticityand visco-elasticity constants measurement apparatus according to claim1, wherein said calculating means carries out during or aftercalculation suitable filtering or moving-averaging of the calculatedelastic constants, visco elastic constants, density, or the reciprocalsif necessary.
 28. The elasticity and visco-elasticity constantsmeasurement apparatus according to claim 1, wherein said calculatingmeans carries out during or after calculation suitable filtering ormoving-averaging of the measured strains, strain rates or accelerationsif necessary particularly at references.
 29. The elasticity andvisco-elasticity constants measurement apparatus according to claim 1,wherein at least one reference is set in an attached reference medium.